Method And Apparatus For Predicting Aggregation Kinetics Of A Biologically Active Material

ABSTRACT

A mechanistic model was developed to extract meaningful thermodynamic and kinetic parameters from an irreversibly denatured process. As a result, methods and computer apparatus have been created that can be used to mathematically determine parameters that are predictive of aggregation kinetics of biologically active materials. Those parameters can then be used to predict stability or aggregation kinetics as a function of time and temperature.

FIELD OF THE INVENTION

The present invention relates to methods and computer products forsimulating and predicting properties of compositions comprisingbiologically active materials. In particular, the invention relates tomethods for rigorously determining kinetic and thermodynamic parametersfor multi-state processes as well as methods for using such parametersto determine, for example, stability and shelf-life.

BACKGROUND OF THE INVENTION

Aggregation of proteins can occur as a consequence of conformationalalterations attributed to denaturation. See, Joly, M. A Physico-ChemicalApproach to the Denaturation of Proteins; Academic Press, Inc., London,1965. In this context, the term “denaturation” refers to “a process (orsequence of processes) in which the conformation of polypeptide chainswithin the molecule are changed from that typical of the native proteinto a more disordered arrangement”. Denaturation can result when aprotein is conformationally perturbed by temperature, pH, or chemicaldenaturants. See, e.g., Remmele, R. L., Jr.; Bhat, S. D.; Phan, D. H.;Gombotz, W. R. Biochemistry 1999, 38, 5241-5247; Kauzmann, W. Somefactors in the interpretation of protein denaturation Adv. Protein.Chem. 1959, 14, 1-64; and Speed-Ricci, M.; Sarkar, C. A.; Fallon, E. M.;Lauffenburger, D. A.; Brems, D. N. Protein Science 2003 12, 1030-1038.

Aside from compromising the integrity of the protein, aggregation canoften lead to decreased solubility and elicit immunogenic responses intherapeutic settings. See, Pinckard, R. N.; Weir, D. M.; McBride, W. H.Clin. Exp Immunol., 1967, 2, 331-341; Moore, W. V.; Leppert, P. J. Clin.Endocrinology Metabolism, 1980, 51, 691-697; and Robbins, D. C.; Cooper,S. M.; Fineberg, S. E.; Mead, P. M. Diabetes, 1987, 36, 838-841. Theimportance in understanding this phenomenon, therefore, has broadimplications not only in the realm of biochemistry, but also in theworld of protein therapeutics.

Low confidence in estimates of the equilibrium parameters that controlthermally irreversible systems has made it difficult to obtainthermodynamically meaningful information about such systems. Althoughthere have been attempts to simplify the reactions, some of theapproximations have depended upon the assumption that k₃>>k₂>>k₁. Thisassumption is only true for completely irreversible systems.

SUMMARY

A more rigorous procedure has been developed for estimating theparameters that have relevance for reversible, partially reversible, andcompletely irreversible processes of thermal denaturing involvingproteins. The derived parameters may apply not only to the thermodynamicaspects of the reactions, but also the kinetic aspects.

The process of protein aggregation can be characterized by thermodynamicand kinetic parameters. See, Krishnamurthy, R.; Manning, M. C. CurrentPharmaceutical Biotechnology, 2002, 3, 361-371; and Grinberg, V. Y.;Burova, T. V.; Haertle, T.; Tolstoguzov, V. B. J. Biotechnology, 2000,79, 269-280. The thermodynamic component characterizes the tendency fora given protein to unfold resulting in a change of state. In many cases,the unfolded state can lead to an irreversibly denatured aggregate statethat is kinetically controlled. See, Vermeer, A. W. P.; Norde, W.Biophys. J., 2000, 78, 394-404 and Sánchez-Riuz, J. M.; López-Lacomba,J. L.; Cortijo, M.; Mateo, P. L. Biochemistry 1988, 27, 1648-1652. Thekinetic component expresses how unfolding contributes to the overallmechanism leading to aggregation (or irreversibly denatured state).

In theory, aggregation is expected to be a second order process andtherefore highly dependent on protein concentration. Moreover, becauseaggregation can involve multiple interactions between two or moremolecules of protein, the aggregation reaction could in some cases beeven greater than second order. Aggregation can also be rate limited bythe formation of an aggregation-competent state that follows first orderreaction kinetics.

The premise for unfolding-mediated aggregation can be explained with theknowledge that protein unfolding typically exposes buried hydrophobicregions of the molecule that become reactive in regard to associationsbetween neighboring molecules that have also unfolded in like manner(via hydrophobic-hydrophobic interactions). See, Brandts, J. F.Thermobiology; Academic Press, 1967, Chapter 3, pp. 25-75. In thisrespect, the change in state is the transition from the compact ornative state to the unfolded or conformationally denatured state. Thekinetics that describe the system can indicate how susceptible theunfolded state is to the interaction between adjacent molecules to formcomplexes. Such systems were recognized by Lumry and Eyring and modeledas three-state systems that involved first an unfolding event followedby an irreversibly denaturing process. See, Lumry, R.; Eyring, H. J.Phys. Chem., 1954, 58, 110-120. The model may be described by the schemebelow,

where N is the native state, U is the unfolded (conformationallydenatured) state and D is the irreversibly denatured state oraggregation product of the reaction. The scheme represented can bedescribed in terms of the kinetics associated with the rates of theforward and reverse reactions (k₁ and k₂) and for the irreversiblydenatured or aggregate state (k₃). The kinetics of the equilibriumbetween the native and unfolded states are also related to thethermodynamics of the reaction since the equilibrium constant may bedescribed as a function of the rates, k₁ and k₂,K ₁₂ =[U]/[N]=k ₁ /k ₂  (1)

Building upon this premise, it would seem that reactions that aredependent upon an unfolding step should exhibit non-Arrhenius profilesproducing curvature in the vicinity of the denaturation temperature (ormelting temperature, T_(m)).

The temperature of denaturation can be determined accurately usingmicrocalorimetry when the system is fully reversible and when the scanrate does not exceed the rate of unfolding. In cases where the system isirreversible, the determination is more complicated and does not lenditself to thermodynamic treatment. Furthermore, if the kinetics of theaggregation process are dependent upon unfolding, a change in kineticbehavior that coincides with the T_(m) of the unfolding transitionshould be apparent. Finally, a microcalorimetric scan rate dependence ofthe T_(m) is expected if the scheme above is applicable and kineticallycontrolled.

One of the central limitations of classic or extended Lumry-Eyringtheory for modeling protein aggregation rates, is the need to generalizereaction mechanisms. Give this limitation, is it possible to extractmeaningful information about the important contributing parameters thatgovern protein aggregation rates? The idea that scan-rate dependentunfolding studies using microcalorimetry could be used to extractkinetic information from reactions that depend upon conformationallyaltered states had been proposed (but not experimentally tested) in the1980's by Privalov and Potekhin. Approaches for extracting meaningfulenthalpies from irreversible microcalorimetric experiments of proteinunfolding has been previously reported. Sanchez-Ruiz and coworkers madethe generalization that in a kinetically controlled process where theintermediate or “U” state was negligibly populated (as in the case forcompletely irreversible protein unfolding reactions where k₃>>k₁), thethree-state Lumry-Eyring model could be simplified to approximate afirst-order reaction from which reliable activation energies thatfollowed Arrhenius behavior could be obtained. Later building upon thiswork, Lepock and coworkers applied a classic three-state Lumry-Eyringmodel to simulate varied rate constant perturbations imposed uponthermodynamic and kinetically controlled steps associated with theunfolding thermogram properties of microcalorimetry data. Finally,Christopher Roberts applied a more general theoretical approach takinginto account 1^(st), 2^(nd), and higher order reaction kinetics ascribedto complex thermodynamic and kinetic properties of irreversibleaggregation reactions in order to predict shelf-life. However, in hisdescription, Roberts had not derived an expression for C_(p) (excessheat capacity) as a function of temperature and scan rate.

In contrast to these approaches, the present work derives a theoreticaltreatment obtained from simulating of scan-rate dependentmicrocalorimetry data in order to extract meaningful thermodynamic andkinetic parameters from a system that is predominantly irreversible andexhibits non-Arrhenius aggregation kinetics. This investigationelucidates the role of thermal unfolding as it pertains to anirreversibly aggregated process involving rhuIL-1R (II). The study ofrhuIL-1R considers the case where aggregate formation results from theassociation of unfolded protein forms and describes protein unfolding asprerequisite through which dimers form, becoming the precursor to allhigher order oligomerized states.

The present invention provides methods and computer products that mightbe used to predict stability of a biologically active material. Incertain embodiments, the stability of the biologically active materialdepends, at least in part, upon a change in the conformational state ofthe material.

According to one embodiment, a composition comprising a biologicallyactive material is provided. The biologically active material is capableof a conformational change due to a thermal change and is substantiallyin its native state prior to such thermal change. A conformationalchange of the biologically active material can be measured as a functionof temperature and time. In certain embodiments, physical or chemicalconsequences of such change are measured. The measurements may be madeunder conditions that result in significant irreversible unfolding, forexample in a predominantly irreversible, scan-rate dependent system.

Various thermodynamic and kinetic parameters can be determined for thebiologically active material based on the measurement(s) of theconformational change and physical or chemical consequences of suchchange. These parameters may be predictive of aggregation kinetics. Incertain embodiments, enthalpy or free energy of transition; ΔC_(p)between native and denatured states of the material; and the temperatureat which about 50% of the protein is in an unfolded state (and about 50%of the protein is in its native state) are determined. The parametersmay collectively model non-Arrhenius aspects of aggregation kinetics.They may also model aggregation as a 2^(nd) order reaction involving twotypes of irreversibly unfolded states.

In certain embodiments of the invention, aggregation kinetics or thestability of the biologically active material can be predicted fromthese thermodynamic and kinetic parameters. For example, in somecircumstances they may be used to predict aggregation kinetics as afunction of time, temperature, and concentration. The parameters mightalso be used to determine different reaction rate constants or beextrapolated to determine shelf life of the biologically active materialat one or more storage temperatures, or to establish a recommendedstorage temperature.

Methods and computer products are disclosed for predicting the effect ofan excipient on shelf life of a biologically active material. One ormore of the kinetic and thermodynamic parameters for the compositioncomprising the biologically active material can be compared to theanalogous parameters for the composition comprising both one or moreexcipients of interest and the biologically active material to determinethe effect of the excipient of interest on the shelf life of thebiologically active material. Or, a property of a biologically activematerial that pertains to aggregation (such as rate or amount ofchemical decomposition, proteolysis, hydrolysis, deamidation, oroxidation) might be predicted.

Briefly, the new approach includes a method for determining parametersfor predicting aggregation kinetics of a biologically active materialcomprising the steps of:

-   (a) providing measurements of conformational change of the    biologically active material at varying temperatures and varying    times, and-   (b) using the measurements of part (a) to mathematically determine    activation energy parameters (E) and frequency factor parameters (A)    associated with at least three different reaction rate constants,    the parameters being predictive of aggregation kinetics of the    biologically active material.

The biologically active material may be a hormone, cytokine,hematopoietic factor, growth factor, antibody, antiobesity factor,trophic factor, anti-inflammatory factor, antibody or enzyme, orerythropoietin, granulocyte-colony stimulating factor, stem cell factor,or leptin. It may be in a formulation that further comprises one or moreexcipients, and steps (a) and (b) may be carried out on a plurality ofdifferent formulations of the material, at least two of the formulationsare at different pH.

The measuring can be carried out using differential scanning calorimetryor size exclusion chromatography. Conformational change of thebiologically active material can be measured as a function oftemperature varied uniformly over time.

Some of the parameters that are determined may collectively modelnon-Arrhenius aspects of the aggregation kinetics. The activation energyparameters (E) and frequency factor parameters (A) may be associatedwith at least four reaction rate constants, or with no more than fourreaction rate constants.

Step (b) can include providing estimated activation energy and frequencyfactor parameters, calculating predicted measurements of conformationalchange based on the estimated parameters, and using an estimation methodto compare predicted measurements to measurements from step (a), forexample with a non-linear least squares fitting method. The approach mayinclude applying a weighting factor dependent on the scan rate. Step (b)can also include may include determining enthalpy or free energy oftransition, determining ΔCp, ΔC_(p) ^(D) ¹ , and ΔC_(p) ^(D) ² (thatchange in heat capacity being predictive of aggregation kinetics of thebiologically active material); or determining the temperature at whichabout 50% of the protein is in an unfolded state and about 50% of theprotein is in its native state. The approach can also involve modelingaggregation, as a function of time at different temperatures, as a firstand second order reaction. It can also include evaluatingidentifiability and variability of one or more of the parameters.

One or more of the following equations is used:{dot over (N)}=−k ₁ N+k ₂ U{dot over (U)}=k ₁ N−(k ₂ +k ₃)U−k ₄ U ²{dot over (D)}=k ₃ U+k ₄ U ²$\begin{matrix}{{C_{P}\left( {v,T} \right)} = {{\left( {{\Delta\quad H_{m}} + {\Delta\quad{C_{P}\left( {T - T_{m}} \right)}}} \right)\left( {{- \frac{1}{v}}N} \right)} + {\Delta\quad C_{P}U} +}} \\{{\frac{k_{3}}{v}\left( {E_{3} + {\Delta\quad{C_{P}^{D_{1}}\left( {T - T_{m}} \right)}}} \right)U} + {\frac{k_{4}}{v}\left( {E_{4} + {C_{P}^{D_{2}}\left( {T - T_{m}} \right)}} \right)U^{2}} +} \\{{\left( {{\Delta\quad C_{P}^{D_{1}}} + {\Delta\quad C_{P}}} \right)D_{1}} + {\left( {{\Delta\quad C_{P}^{D_{2}}} + {\Delta\quad C_{P}}} \right)D_{2}}}\end{matrix}$A _(gg)(T,t)=D.

Once derived, the parameters can be used to predict stability oraggregation kinetics as a function of time and temperature using atleast three different reaction rate constants. For example, the level ofaggregation of the biologically active material as a function oftemperature, time, and concentration of said biologically activematerial might be determined, or might be determined at a giventemperature, for example at a temperature of 40 degrees C. or less, orin a range from 4 to 25 degrees C., or from 15 to 30 degrees C., or from−5 to 15 degrees C., or from 2 to 8 degrees C. The level of aggregationmight be predicted after a time period, for example a time of six monthsor more, or one year or more, or two years or more. Or, it may involvepredicting time to reach an unacceptable level of aggregation, such asthe time to reach 50% aggregation. It might also involve predictingaggregation half-life of the biologically active material (with orwithout incipients) as a function of temperature and concentration ofthe biologically active material, predicting shelf-life of the materialat one or more storage temperatures, or predicting an optimal storagetemperature. It might also be used to predict stability or level ofaggregation for a plurality of formulations of the material, or forformulations that contain one or more excipients.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is an illustration of unnormalized SEC results of rhuIL-1R(II) at58° C. showing the progression of aggregates over time in seconds.Additionally, the arrows indicate the changes in the monomer (downward)and the aggregate (upward). The vertical lines bracketing elution timesbetween 5.7 minutes and 8.4 minutes represent the integration regiondescribing total aggregation. The eluting component near 8 minutes isassigned to the dimer population. It is relatively stable at differenttemperatures and time, thus supporting the pseudo-steady-stateaggregation mechanism.

FIG. 2 is a pair of proposed enthalpy and free energy diagramsdescribing the thermal unfolding and aggregation of rhuIL1R(II). Nrepresents the native state; U, the unfolded state; D1, aggregatesformed through 1^(st) order reactions; and D2, aggregates from 2^(nd)order reactions. D₁* and D₂* are the corresponding transition statesleading to aggregation. Activation free energies related to each stateare denoted as ΔG_(i) ^(‡) (where i=1, 2, 3, or 4.) The free energy ofunfolding is ΔG₁₂ and aggregation free energy is ΔG_(agg). Astemperature approaches Tm, ΔG₁₂ goes to zero.

FIG. 3 is a comparison of the calculated result using eqn (31) (solidlines) with experimental data (broken lines) for the DSC scan ratedependent experiment. From left to right, the scanning rates are 0.25,0.5, 1.0, and 1.5° C./min, respectively. The inset figure is a DSC scanfor a 91% reversible rhuIL-1R(II) system in 0.1 M sodium phosphate plus2 M urea at pH=7.0. In all data shown the transition baseline has beensubtracted.

FIG. 4 is a pair of charts of the time-temperature data of aggregation.The predicted fits based upon the model (eqn (20)) are depicted by thesolid lines. Data points represented by the assorted symbols are theexperimentally determined values. In the top figure, the temperaturesfrom top to bottom are 75° C., 69° C., 65° C., 58° C., and 50° C. In thebottom figure, the temperatures from top to bottom are 39° C., 37° C.,and 34° C.

FIG. 5 is a set of van't Hoff plots of data for rhuIL1R(II). The topplot (A) shows the best least squares fit of data to two distinctlydifferent lines showing least squares equation for pre and postunfolding transition temperature zones. The cross over point of the twolines is in the vicinity of the T_(m). The bottom plot (B) compares theinitial aggregation rates with * being the calculation based on themodel (eqn (21)); the dashed line being the rate approximation$\left( {k^{\prime} = \frac{k_{1}k_{3}}{k_{1} + k_{2} + k_{3}}} \right);$and ∘ being the experimental data. The experimental error in temperatureis indicated by ‘+’ on either side of the data points. The data denotedby ● is the Sánchez-Ruiz et al. treatment of the scan rate data.

FIG. 6 is a plot of the temperature dependence of the kinetic rateconstants for 2 mg/mL rhuIL-1R(II). The point denoting the temperaturewhere k₁=k₂ is the extracted T_(m).

FIG. 7 is a comparison of the concentration dependence predicted by themodel using the parameters extracted. The curves depict the behaviorpredicted by the model and the data points represent measuredaggregation by SEC. The upper curve and points correspond to data at67.5° C. for a duration of 2 minutes. The lower ones correspond to dataat 42.5° C. for 48 hours.

FIG. 8 is a flowchart of the method used in one embodiment of theinvention.

FIG. 9 is a flowchart of a method of evaluating identifiability ofparameters obtained using the model.

FIG. 10 is a flow chart of a method of evaluating variability ofparameters obtained using the model

FIG. 11 is a block diagram of a computer system that can be used toimplement various aspects of this invention.

DETAILED DESCRIPTION

Abbreviations and definitions will be provided before an overview andspecific applications are discussed.

I. Abbreviations and Definition

The following abbreviations and definitions will be used.

A. Abbreviations

The following abbreviations are used:

T=temperature (in K unless otherwise noted)

t=time (in minutes unless otherwise noted)

G=Gibbs free energy (in kcal/mol)

S=entropy (in kcal/mol/K)

H=enthalpy (in kcal/mol)

h=enthalpy of the ensemble state (in kcal/mol)

E=activation energy

ε=energy of the ensemble state

K_(B)=the Bolzmann constant

R=the gas constant (0.0019872 kcal/mol/K)

DSC=differential scanning calorimetry

SEC=size exclusion chromatography

CD=circular dichroism

STP=standard temperature and pressure

Subscripts and superscripts are applied to indicate state.

This invention is not limited to particular methods, devices, orsystems, which can, of course, vary. The terminology used herein is forthe purpose of describing particular embodiments only, and is notintended to be limiting.

As used in this specification and the appended claims, the singularforms “a,” “an,” and “the” include plural referents unless the contentclearly dictates otherwise. Thus, for example, reference to “a surface”includes a combination of two or more surfaces; reference to “protein”can include mixtures of protein, and the like.

Unless defined otherwise, all technical and scientific terms have thesame meaning commonly understood by one of ordinary skill in the art towhich the invention pertains.

B. Definitions

“Aggregation” refers to one of the most common protein degradationpathways which results in tangled and amorphous masses of proteinfibers.

“Biologically active material” refers to a molecule that is capable of aconformational change and, in certain embodiments, unfolding, due to athermal change.

“Denaturation” refers to a process in which the conformation of amolecule is changed from a first state to a more disordered arrangement.In a certain embodiment wherein the molecule is a protein, denaturationrefers to a process (or sequence of processes) in which the conformationof polypeptide chains with the protein are changed from that typical ofthe native protein to a more disordered arrangement without cleavage ofany of the primary chemical bonds that link one amino acid to another.Cleland et al. Critical Reviews in Therapeutic Drug Carrier Systems10(4): 307-377 (1993).

“Denatured nucleic acid” refers to a nucleic acid that has been treatedto remove folded, coiled, or twisted structure. Denaturation of atriple-stranded nucleic acid complex is complete when the third strandhas been removed from the two complementary strands. Denaturation of adouble-stranded DNA is complete when the base pairing between the twocomplementary strands has been interrupted and has resulted insingle-stranded DNA molecules that have assumed a random form.Denaturation of single-stranded RNA is complete when intramolecularhydrogen bonds have been interrupted and the RNA has assumed a random,non-hydrogen bonded form.

“Denatured protein” refers to a protein which has been treated to removesecondary, tertiary, or quaternary structure.

“Differential scanning calorimetry” refers to an analytical method basedupon the detection of changes in the heat content or the specific heatof a sample with temperature.

“Evaluating identifiability” of parameters refers to evaluating theuniqueness of the parameters. For example, a first set of parameters isnot considered identifiable if a different second set of parameters canresult in the same predicted measurements of conformational change thatwould be predicted by the first set of parameters. One embodiment of amethod of evaluating identifiability of parameters is illustrated inFIG. 9.

“Evaluating variability” of parameters refers to evaluating howvariability in actual measurements affects the parameters determinedfrom these measurements of conformational change. In such an evaluation,it is assumed that the parameters are identifiable. For example, addingrandom variability (i.e. “noise”) to the measurements should result insome but relatively little change in the determined parameters; fromthis, it is possible to determine a 95% confidence interval for a rangeof parameter values. One embodiment of a method of evaluatingvariability of parameters is illustrated in FIG. 10.

“Folding,” “refolding,” and “renaturing” refer to the acquisition of thecorrect secondary, tertiary, or quaternary structure, of a protein or anucleic acid, which affords the full chemical and biological function ofthe biomolecule.

“Glass,” “glassy state,” or “glassy matrix,” refers to a liquid that haslost its ability to flow, i.e. it is a liquid with a very highviscosity, wherein the viscosity ranges from 10¹⁰ to 10¹⁴pascal-seconds. It can be viewed as a metastable amorphous system inwhich the molecules have vibrational motion but have very slow (almostimmeasurable) rotational and translational components. As a metastablesystem, it is stable for long periods of time when stored well below theglass transition temperature. Because glasses are not in a state ofthermodynamic equilibrium, glasses stored at temperatures at or near theglass transition temperature relax to equilibrium and lose their highviscosity. The resultant rubbery or syrupy, flowing liquid is oftenchemically and structurally destabilized.

The “glass transition temperature” is represented by the symbol T_(g)and is the temperature at which a composition changes from a glassy orvitreous state to a syrup or rubbery state. Generally, T_(g) isdetermined using differential scanning calorimetry and is standardlytaken as the temperature at which onset of the change of heat capacity(Cp) of the composition occurs upon scanning through the transition.There is no present international convention for the definition ofT_(g). The T_(g) can be defined as the onset, midpoint, or endpoint ofthe transition; for purposes of this invention, the midpoint of thetransition will be used. See the article by C. A. Angell, Science, 267,1924-1935 (1995) and Jan P. Wolanczyk, Ciyo-Letters, 10, 73-76 (1989).For detailed mathematical treatment see Gibbs and DiMarzio, Journal ofChemical Physics, 28, No. 3, 373-383 (March, 1958). These articles areincorporated herein by reference.

A “higher stability” is indicated by a lower aggregation rate or alonger shelf-life. An alternative exemplary measure of stability is thefree energy (ΔG) of unfolding from a native state to a denatured state.A higher free energy indicates a more stable biologically activematerial. For example, a typical of ΔG for proteins is usually 5-25kcal/mol, but a given ΔG for a particular protein can be increased byusing a stabilizing formulation of biologically active material, e.g. byadding excipients, or adjusting pH or concentration of the biologicallyactive material. “Lyophilize” or “lyophilization” or “freeze-dry” willrefer to a process for the removal of water from frozen compositions bysublimation under reduced pressure.

“Measuring” a conformational change will include measuring the changedirectly or measuring the change indirectly by measuring a physical orchemical consequence of such change.

The “midpoint temperature” or “T_(m)” is the temperature midpoint of athermal denaturation curve. The T_(m) can be readily determined usingmethods well known to those skilled in the art. See, for example, Weber,P. C. et al., J. Am. Chem. Soc. 116:2717-2724 (1994); Clegg, R. M. etal., Proc. Nati. Acad. Sci. U.S.A. 90:2994-2998 (1993).

“Native protein” refers to a protein which possesses the degree ofsecondary, tertiary, or quaternary structure that provides the proteinwith full chemical and biological function.

“Nucleic acid,” “oligonucleotide,” or grammatical equivalents hereinmean a molecule comprising at least ten nucleotides covalently linkedtogether. A nucleic acid will generally contain phosphodiester bonds,although in some cases, as outlined below, nucleic acid analogs areincluded that may have alternate backbones, comprising, for example,phosphoramide (Beaucage et al., Tetrahedron 49(10):1925 (1993) andreferences therein; Letsinger, J. Org. Chem. 35:3800 (1970); Sprinzl etal., Eur. J. Biochein. 81:579 (1977); Letsinger et al., Nucl. Acids Res.14:3487 (1986); Sawai et al, Chem. Lett. 805 (1984), Letsinger et al.,J. Am. Chem. Soc. 110:4470 (1988); and Pauwels et al., Chemica Scripta26:141 91986)), phosphorothioate (Mag et al., Nucleic Acids Res. 19:1437(1991); and U.S. Pat. No. 5,644,048), phosphorodithioate (Briu et al.,J. Am. Chem. Soc. 111:2321 (1989), O-methylphophoroamidite linkages (seeEckstein, Oligonucleotides and Analogues: A Practical Approach, OxfordUniversity Press), and peptide nucleic acid backbones and linkages (seeEgholm, J. Am. Chem. Soc. 114:1895 (1992); Meier et al., Chem. Int. Ed.EngI. 31:1008 (1992); Nielsen, Nature, 365:566 (1993); Carlsson et al.,Nature 380:207 (1996), all of which are incorporated by reference).Other analog nucleic acids include those with positive backbones (Denpcyet al., Proc. Natl. Acad. Sci. USA 92:6097 (1995); non-ionic backbones(U.S. Pat. Nos. 5,386,023, 5,637,684, 5,602,240, 5,216,141 and4,469,863; Kiedrowshi et al., Angew. Chem. Intl. Ed. English 30:423(1991); Letsinger et al., J. Am. Chem. Soc. 110:4470 (1988); Letsingeret al., Nucleoside & Nucleotide 13:1597 (1994); Chapters 2 and 3, ASCSymposium Series 580, “Carbohydrate Modifications in AntisenseResearch,” Ed. Y. S. Sanghui and P. Dan Cook; Mesmaeker et al.,Bioorganic & Medicinal Chem. Lett. 4:395 (1994); Jeffs et al., J.Biomolecular NMR 34:17 (1994); Tetrahedron Lett. 37:743 (1996)) andnon-ribose backbones, including those described in U.S. Pat. Nos.5,235,033 and 5,034,506, and Chapters 6 and 7, ASC Symposium Series 580,“Carbohydrate Modifications in Antisense Research,” Ed. Y. S. Sanghuiand P. Dan Cook. Nucleic acids containing one or more carbocyclic sugarsare also included within the definition of nucleic acids (see Jenkins etal., Chem. Soc. Rev. (1995) pp 169-176).

An “optimal formulation” is a formulation, including, for example,pharmaceutically acceptable excipients, pH and concentration ofbiologically active material, that exhibits higher stability relative toother formulations.

An “optimal storage temperature” is a recommended storage temperaturerange that provides higher stability relative to other typical storagetemperatures. For example, refrigeration at 0-8 degrees C. may providehigher stability for the biologically active material relative to roomtemperature storage at 21-23 degrees C. “Pharmaceutically acceptable”excipients (vehicles, additives) are those which can reasonably beadministered to a subject mammal to provide an effective dose of theactive ingredient employed. Preferably, these are excipients that theFederal Drug Administration (FDA) has to date designated as GenerallyRegarded as Safe (GRAS).

“Pharmaceutical composition” refers to preparations that are in such aform as to permit the biological activity of the active ingredients tobe unequivocally effective, and that contain no additional componentsthat are toxic as administered to the subjects.

The terms “polypeptide,” “peptide,” and “protein” are usedinterchangeably herein to refer to a polymer of amino acid residues, andincludes peptides, polypeptides, consensus molecules, antibodies,analogs, derivatives, or combinations thereof. The terms apply to aminoacid polymers in which one or more amino acid residues is an artificialchemical analogue of a corresponding naturally occurring amino acid, aswell as to naturally occurring amino acid polymers. Amino acids may bereferred to herein by either their commonly known three letter symbolsor by Nomenclature Commission. Nucleotides, likewise, may be referred toby their commonly accepted single-letter codes, i.e., the one-lettersymbols recommended by the IUPAC-IUB. “Predicting stability” refers tothe prediction of the ability of a composition comprising a biologicallyactive material to retain its physical stability, chemical stability,and/or biological stability upon storage. Prediction encompasses thedetermination of a recommended storage temperature; the determination ofshelf-life at one or more storage temperatures, including therecommended storage temperature; the determination of the amount or rateof aggregation of the biologically active material for temperatures andtime periods of interest; and the determination of other parameters aswill be appreciated by one of skill in the art.

“Predicting stability” refers to the prediction of the ability of acomposition comprising a biologically active material to retain itsphysical stability, chemical stability, and/or biological stability uponstorage. Prediction encompasses the determination of a recommendedstorage temperature; the determination of shelf-life at one or morestorage temperatures, including the recommended storage temperature; thedetermination of the amount or rate of aggregation of the biologicallyactive material for temperatures and time periods of interest; and thedetermination of other parameters as will be appreciated by one of skillin the art.

“Recommended storage temperature” for a composition is the temperature(T) at which a drug composition is to be stored to maintain thestability of the drug over the shelf life of the composition in order toensure a consistently delivered dose. This temperature is initiallydetermined by the manufacturer of the composition and approved by thegovernmental agency responsible for approval the composition formarketing (e.g., the Food and Drug Administration in the U.S.). Thistemperature will vary for each approved drug depending on thetemperature sensitivity of the active drug and other materials in theproduct. The recommended storage temperature will vary from about −70°C. to about 40° C. but powdered and liquid drug compositions aregenerally recommended for storage between about 4° C. and about 25° C.Usually a drug will be kept at a temperature that is at or below therecommended storage temperature.

A biologically active material “retains its biological activity” in apharmaceutical composition if the biological activity of thebiologically active material at a given time can be within about 10%(within the errors of the assay) of the biological activity exhibited atthe time the pharmaceutical composition was prepared as determined in abinding assay, for example.

A biologically active material “retains its chemical stability” in apharmaceutical composition if the chemical stability at a given time issuch that the biologically active material is considered to still retainits biological activity as defined above. Chemical stability can beassessed by detecting and quantifying chemically altered forms of thebiologically active material. Chemical alteration may involve sizemodification (e.g. clipping of proteins) which can be evaluated usingsize exclusion chromatography, SDS-PAGE and/or matrix-assisted laserdesorption ionization/time-of-flight mass spectrometry (MALDI/TOF MS),for example. Other types of chemical alteration include chargealteration (e.g. occurring as a result of deamidation) which can beevaluated by ion-exchange chromatography, for example.

A biologically active material “retains its physical stability” in apharmaceutical composition if, e.g., aggregation, precipitation, and/ordenaturation upon visual examination of color and/or clarity, or asmeasured by UV light scattering or by size exclusion chromatography, arenot significantly changed.

“Size exclusion chromatography” refers to a separation technique inwhich separation mainly according to the hydrodynamic volume of themolecules or particles takes place in a porous non-adsorbing materialwith pores of approximately the same size as the effective dimensions insolution of the molecules to be separated

A “stable” formulation or composition is one in which the biologicallyactive material therein essentially retains its physical stability,chemical stability, and/or biological activity upon storage. Variousanalytical techniques for measuring stability are available in the artand are reviewed, e.g., in Peptide and Protein Drug Delivery, 247-301,Vincent Lee Ed., Marcel Dekker, Inc., New York, N.Y., Pubs. (1991) andJones, A. Adv. Drug Delivery Rev. 10: 29-90 (1993). Stability can bemeasured at a selected temperature for a selected time period.

“Shelf stability” or “shelf life” refers to the loss of specificactivity and/or changes in secondary structure from the biologicallyactive material over time incubated under specified conditions.

In a pharmacological sense, a “therapeutically effective amount” of abiologically active material refers to an amount effective in theprevention or treatment of a disorder wherein a “disorder” is anycondition that would benefit from treatment with the biologically activematerial. This includes chronic and acute disorders or diseasesincluding those pathological conditions which predispose a patient tothe disorder in question.

A “thermal denaturation curve” is a plot of the physical changeassociated with the denaturation of a protein or a nucleic acid as afunction of temperature. See, for example, Davidson et al, NatureStructure Biology 2:859 (1995); Clegg, R. M. et al., Proc. Natl. Acad.Sci. U.S.A. 90:2994-2998 (1993).

“Two-state process” refers to a process wherein the measuredcalorimetric enthalpy change is equivalent to the effective two-statevan't Hoff enthalpy change. A “multi-state process” includes a two-stateprocess and higher order processes.

II. Overview

The present invention provides methods for simulating multi-stateprocesses of instability of proteins and other biologically activemolecules. More specifically, when instability depends upon a change inconformational state of the molecule, the invention may enable oneskilled in the art to rigorously determine kinetic and thermodynamicparameters related to that process. These parameters might then be usedto predict an aggregation property (such as shelf-life) of thebiologically active material, including for example, the effect ofconcentration, pH, or excipients on shelf-life.

This overview includes a brief description of materials and methods thatwere used in developing the invention, and is followed by a theoreticaltreatment.

A. Materials and Methods

The materials and methods that were used in developing the invention aredescribed below.

1. Material

Purified rhuIL-1R(II) (recombinant human Interleukin-1 receptor, typeII) was obtained as a bulk drug concentrate (˜10 mg/mL) in a phosphatebuffered saline solution (PBS: 20 mM sodium phosphate (pH 7.4), 150 mMNaCl) obtained from Immunex Corporation (now Amgen, Inc.). The protein,expressed in CHO cells was approximately 20% glycosylated. Proteinconcentrations were determined spectrophotometrically at 280 nm using anexperimentally determined molar extinction coefficient of 1.61 mL/mg-cm.All excipients used were reagent grade or better. The proteinpolypeptide molecular weight was approximately 38 kD.

2. Methods

Two studies were used in developing this invention, and are describedmore fully below.

a. Microcalorimetry (DSC Experiment)

Samples were evaluated in a vp-DSC (MicroCal, Inc.) using scan rates of0.25, 0.5, 1.0 and 1.5° C./min. Protein solutions were fixed at 2 mg/mL(unless otherwise noted) by diluting with the PBS buffered solution. TheT_(m) dependence on scan rate was assessed in the microcalorimeter usingthe method described by Sanchez-Ruiz and coworkers.

Thermal reversibility of a 2 mg/mL solution was also examined in PBS ata scan rate of 1° C./min within the time frame of differential heatingto 90° C., followed by cooling, re-equilibrating, and subsequentlyreheating a second time (time lapse between scans was essentially 1 hr).The data were evaluated using Origin software (version 5.0) providedwith the instrument.

b. Time-Temperature Aggregation Studies

The temperatures selected for the time-temperature aggregation studiescovered a broad range that straddled the unfolding transition endotherm.The main idea was to traverse the transition region that includedtemperatures well outside the transition envelope above and below theapparent T_(m) (˜58° C. at a scan rate of about 1° C./min).

The kinetics of the aggregation reaction were studied by placing 0.5 mLof protein solution in a 2 mL capacity polypropylene eppendorf vial andheating the contents at a designated temperature in an appropriateheating device (either incubator or heating block) for a designatedperiod of time. For studies conducted using a heating block, carefulattention was given to temperature control (±1° C.) and uniform heatingof the sample. Bored wells (1-cm inside diameter) in an aluminum blockwere filled with water and allowed to equilibrate at the desiredtemperature prior to insertion of the eppendorf vial. At elevatedtemperatures (>T_(m)) equilibrium was reached immediately. In the lowtemperature studies (<40° C.), heating experiments were carried out inthe incubator and given adequate time (2 hrs) to reach the equilibriumtemperature prior to starting the clock. This was established by directmonitoring of the sample when temperature exhibited no greater changethan±1° C. at equilibrium with the surrounding environment. Atdesignated time points samples were removed from the heating device,immediately placed on ice, and stored in the refrigerator beforeexamination by size exclusion chromatography (SEC).

Analysis was carried out on a HP-1100 HPLC system. Samples were elutedoff a TosoHaas TSK-G3000 SWXL column at 1 mL/min with 100 mM phosphate(pH 6.5), 50 mM NaCl eluent. A 20 μg sample injection load was used perHPLC run. The kinetics were determined by assessing the total amount ofaggregation (expressed as a percentage of the total area under thesample protein peaks) at a designated time, as shown in FIG. 1 for the58° C. data as an example. The region of integration was defined by thevertical lines bracketing the elution times extending from about 5.7 to8.4 minutes (FIG. 1). All aggregation measurements were determined inthe same way at other temperatures studied.

Attention to possible competing side reactions (i.e., breakdown) wasinvestigated and found to be negligible (no evidence) throughout thetime duration of the studies presented. Hence, one could be assured thatthe aggregation pathway was the primary instability being detectedduring the experiments. Furthermore, there was no evidence of proteininsolubility in all cases studied herein.

Detection of the eluting components was achieved with a photodiode arraydetector monitoring absorbance at 220 nm. The main peak eluting near 9minutes and the peak eluting at 8 minutes were confirmed to be monomerusing the “three detector” light scattering method described previously.See, Wen, J.; Arakava, T.; Talvenheimo, J.; Weicher, A. A.; Horan, T.;Kita, Y.; Tseng, J.; Nicolson, M.; Philo, J. S. Techniques in ProteinChemistry 1996, VII, pp23-31 and Wen, J.; Arakava, T.; Philo, J. S.Analytical Biochemistry 1996, 240, 155-166. It should be noted that theSEC aggregation result is assumed to accurately reflect solution statecomposition.

B. Theoretical Treatment

The goal of the theoretical modeling in this work was to describe thedominant underlying physical processes on a macroscopic level, and toextract as accurately as possible the kinetic and thermodynamicparameters. Because many of the parameters are related to each other, itis important to incorporate these relationships in the model and to usesome experimentally derived quantities to cross check the consistency.

In the next section, a description of the system is followed bydiscussions of the pertinent kinetic equations and reaction rates, andthe observables that were used when developing the approach.

1. System Description

Previous published work pertaining to the fitting or simulation of DSCexperiments were described by kinetic and thermodynamic contributions.Kinetic equations describe the rate of unfolding and aggregationreactions while the measured thermodynamic quantity is the excess heatcapacity C_(P). In the case of a two-state reversible system in asteady-state (see FIG. 2, where N is the average of the ensemble nativestate, and U is the average of the ensemble unfolded state), thethermodynamic part can be simply described by an equilibrium constant ofthe unfolding reaction, $\begin{matrix}\begin{matrix}{K_{12} = {\exp\left( {{- \Delta}\quad{G/({RT})}} \right)}} \\{= {\exp\left( {{\Delta\quad{S/R}} - {\Delta\quad{H/({RT})}}} \right)}}\end{matrix} & (2)\end{matrix}$where ΔG, ΔS, and ΔH correspond to the change in Gibbs free energy,entropy, and enthalpy pertaining to the reaction. It is important tonote that the thermodynamic parameter, K₁₂, can also be described interms of the kinetic rate constants for the forward and reversereactions as described in eqn (1). The measured quantity in the DSCexperiment is the excess heat capacity,${C_{P}(T)} = {{\left( {{\Delta\quad H_{m}} + {\Delta\quad{C_{P}\left( {T - T_{m}} \right)}}} \right)\left( {- \frac{\mathbb{d}N}{\mathbb{d}T}} \right)} + {\Delta\quad{C_{P} \cdot {U(T)}}}}$where the temperature increases linearly in time. ΔC_(P) is the changein heat capacity between the native and the denatured states. It hasbeen ascribed to the exposure of hydrophobic surface to the solventduring thermal unfolding. Temperature may be described in terms of thescan rate ν (° C./min) in the expression T=T₀+νt where T₀ is the initialtemperature of the scan.

For a fully reversible system, thermodynamically meaningful parameterscan be determined. In regard to rhuIL-1R(II), a fully reversiblecalorimetry experiment was nearly achieved when the proteinconcentration was 0.44 mg/mL in a solution consisting of 2 M urea(nondenaturing by CD at 20° C.) and 0.1 M sodium phosphate for bufferingat pH 7 (see inlay of FIG. 3). The thermodynamic quantities measured inthe fully reversible case are ΔHcal=82.5±2.5 kcal/mol, ΔC_(p)=1.0±0.5kcal/mol/K, and T_(m)=325.8±0.2° K. In this case the ΔH_(cal)/ΔH_(vh)ratio was ˜0.9, (where ΔH_(cal) is the calorimetric and/ΔH_(vh) is thevan't Hoff enthalpies) suggesting the unfolding process was essentiallytwo-state. Although urea can shift the Tm to lower temperatures andlower the enthalpy of unfolding, we have used non-denaturing levels ofurea (as measured by CD at 20° C.) that should minimally perturb theunfolding transition, allowing the conditions used to represent anapproximate reference point of the true T_(m) and associatedthermodynamic parameters in the absence of the irreversible step.

The experiment showed that urea effectively blocked the progress ofaggregation, and that the system achieved 91% thermal reversibility inits presence. In the absence of urea, massive aggregation was observedat T>T_(m) (see FIG. 4), and the process was found to be predominantlyirreversible. The DSC scan experiment also produced different behaviors,the most prominent being an increase in the apparent melting temperaturewith increasing scan rates in addition to a possibly negative influencein the ΔC_(p) on the high temperature side of the unfolding envelope. Anexplanation for this phenomenon affiliated with aggregation has beenreported previously.

Another important observation between the fully reversible and thepartially irreversible thermal denaturation studies of rhuIL-1R(II)carried out in the DSC is the additional heat of the reaction making thetotal AH quantity higher than the urea experiment. This amounts to ˜48kcal/mol more heat in the irreversible reaction approaching a total ΔHof 130 kcal/mol (FIG. 3). This would indicate that an additionalendothermic contribution exists within the reaction process thatinvolves the aggregate. It is proposed that this additional heat mayarise from subsequent unfolding contributions resulting fromprotein-protein adsorption leading to the two D aggregated states of themodel (as depicted in FIG. 2). Such endothermic heats have been observedfor bovine milk a-lactalbumin where the subprocesses of sorbent(negatively-charged polystyrene latex), protein dehydration and proteindenaturation contribute to the overall driving force of surfaceadsorption. In the case of rhuIL-1R(II), the sorbent is either anotherunfolded protein or soluble aggregate, and dehydration occurs by theremoval of water from hydrophobic interfaces with the solvent, drivingthe reaction to the D states. In this process, a favorable increase inthe entropy of the solution is expected resulting from hydrophobicsurface area reduction as the aggregates continue to grow. In otherwords, the aggregation reaction of unfolded protein molecules in aqueoussolution is an entropically driven reaction.

As for kinetics, the SEC data shown in FIG. 4 clearly indicate thatthere is a transition temperature where the aggregation rate of thesystem changes so that one cannot simply explain the denaturationprocess in terms of a single step described by an Arrhenius rate as inthe work of Sánchez-Ruiz and coworkers. Nevertheless it is still auseful model to obtain an estimated activation energy. The expandedmodel described by Lepock et al. appears to be a better description ofthe system where all reactions are modeled as first order reactions.

However, our experimental observation of the concentration dependence ofaggregation revealed that the approximate order of reaction was1.70±0.04, suggesting a mixture of first and second order reactionsparticipate in the system. Therefore, we add to Lepock's model an extraterm describing the contribution of the second order processes. Morespecifically, we denote the aggregates resulting from the first orderterm as D₁, and those from the second order term as D₂. Together theyare the irreversibly denatured population, D, with D₁* and D₂*representing the corresponding transition states that conceivably areaggregation competent species (see FIG. 2). The reaction can berepresented by

In the pathway from U

D₁* or U

D₂*, the reaction is assumed ₁ ^(st) order and reversible. From D₁*→D₁and from D₂*→D₂ the reaction follows 2^(nd) order kinetics. Within the U

D₁*→D₁ path, the U→D₁* is rate limited and therefore, the overallreaction from U→D₁ is 1^(st) order. Likewise, in the path from U

D₂*→D₂, the D₂*→D₂ step is rate limiting and therefore the overallkinetics from U→D₂ is second order.

It should be noted in FIG. 4 that at the elevated temperature above 58°C. the reaction does not approach 100% aggregate. The data indicate apoint of saturation that appears to approach 84% to 87%. Thisobservation indicates that there is a remnant of protein that does notparticipate in the aggregation process. This remnant contribution isalso observed in DSC thermal reversibility experiments and suggests thatthe approximate 13% of rhuIL-1R(II) can be accounted as fully reversiblespecies in contrast to the majority of the molecules that go through theunfolding process leading to aggregates.

2. The Kinetic Equations and the Reaction Rates

Since the quantity or population of the transition state (i.e., D₁* andD₂*) are short lived, leading rapidly to the final states, D₁ and D₂,the kinetics of the system may be represented by

Assuming the total molar concentration of protein in the solution is[N₀], the molar concentration at each state can be normalized by thisnumber to get a dimensionless relative concentration, i.e.,$\begin{matrix}{N = \frac{\lbrack N\rbrack}{\left\lbrack N_{0} \right\rbrack}} & (3) \\{U = \frac{\lbrack U\rbrack}{\left\lbrack N_{0} \right\rbrack}} & (4) \\{D = \frac{\lbrack D\rbrack}{\left\lbrack N_{0} \right\rbrack}} & (5)\end{matrix}$

It is important to realize that [D] represents the molar concentrationof irreversibly denatured protein (aggregate weightconcentration/monomer protein molecular weight). It is a collection ofaggregate expressed as an equivalent portion of monomeric molecularforms.

The initial condition of the system is N(t=0)=1, U (t=0)=0 and D(t=0)=0.The differential equations describing the kinetics of the system are asfollows:{dot over (N)}=−k ₁ N+k ₂ U  (6){dot over (U)}=k ₁ N−(k ₂ +k ₃)U−k ₄ U ²  (7){dot over (D)}=k ₃ U+k ₄ U ²  (8)where the U term results from the second order process and where D=D₁+D₂with D₁ corresponding to the first order term physically representingthe reaction of the unfolded molecule with a constant supply ofaggregate molecules to form adducts and D₂ corresponds to the secondorder term depicting the formation of pseudo-steady-state dimmer leadingto higher order aggregates. The terms k₁, k₂, k₃, and k₄ are thecorresponding rate constants.

The second order reaction coefficient k₄ is necessarily proportional tothe concentration [N₀], so that equations (6)-(8) hold for any [N₀]. Thefollowing derivation illustrates this conclusion: Eqn (7) is derivedfrom the kinetic equation for the unnormalized concentration [U]:[{dot over (U)}]=k ₁ [N]−(k ₂ +k ₃)[U]−k ₄ ′[U] ²  (9)Dividing eqn (9) by [N₀] on both sides one obtains $\begin{matrix}{\frac{\left\lbrack \overset{.}{U} \right\rbrack}{\left\lbrack N_{0} \right\rbrack} = {{{k_{1}\frac{\lbrack N\rbrack}{\left\lbrack N_{0} \right\rbrack}} - {\left( {k_{2} + k_{3}} \right)\frac{\lbrack U\rbrack}{\left\lbrack N_{0} \right\rbrack}}} = {{k_{4}^{\prime}\left\lbrack N_{0} \right\rbrack}{\left( \frac{\lbrack U\rbrack}{\left\lbrack N_{0} \right\rbrack} \right)^{2}.}}}} & (10)\end{matrix}$Now by substituting in the definitions of eqns (3)-(5) one arrives at{dot over (U)}=k ₁ N−(k ₂ +k ₃)U−(k ₄ ′[N ₀ ]) U ².  (11)In order for eqn (11) to agree with eqn (7), it must be thatk₄=k₄′[N₀],so that k₄′ denotes a linear rate proportionality constant.

The linear dependence of k₃ on [N₀] is not an obvious because it appearsas a coefficient for a first order reaction rate. Since it originatesfrom the reaction of one unfolded molecule with any aggregate, it isreasonable to assume the same linear relation to [N₀] holds true, andtherefore both k₃ and k₄ can be expressed ask ₃ =k ₃ ′[N ₀]  (12)k ₄ =k ₄ ′[N ₀]  (13)k₃′ denotes a linear rate proportionality constant analogous to k₄′. Itis important to recognize that k₃ is proportional to the initial proteinconcentration.

The unfolding rate coefficient k₁ is known to be well described by theArrhenius law:k ₁=exp (A ₁ −E ₁ /RT).  (14)Recalling the experiment with urea (see the inlay of FIG. 3), a positivechange in the baseline molar heat capacity, ΔC_(P), was observed. Thisimplies that k₂ has non-Arrhenius behavior (see eqn (17)), as derivedfrom the equilibrium constant of the folding/unfolding reaction and themodified Gibbs-Helmholz equation: $\begin{matrix}{{\Delta\quad G_{12}} = {{\Delta\quad{H_{m}\left( {1 - \frac{T}{T_{m}}} \right)}} + {\Delta\quad{C_{P}\left\lbrack {T - T_{m} - {T\quad{\ln\left( \frac{T}{T_{m}} \right)}}} \right\rbrack}}}} & (15)\end{matrix}$

From this, one can write the corresponding equilibrium constant as$\begin{matrix}\begin{matrix}{K_{12} = \frac{k_{1}}{k_{2}}} \\{= {\exp\left( {{- \Delta}\quad{G_{12}/({RT})}} \right)}} \\{= {\exp\left( {{{- \Delta}\quad{H_{m}\left( {1 - \frac{T}{T_{m}}} \right)}} + {\Delta\quad{{C_{P}\left\lbrack {T - T_{m} - {T\quad{\ln\left( \frac{T}{T_{m}} \right)}}} \right\rbrack}/({RT})}}} \right)}}\end{matrix} & (16)\end{matrix}$Substituting eqn (14) into eqn (16) and defining E₁-E₂≡ΔH_(m), and${A_{1} - A_{2}} \equiv \frac{\Delta\quad H_{m}}{{RT}_{m}}$$\left( {{{therefore}\quad T_{m}} = \frac{E_{1} - E_{2}}{R\left( {A_{1} - A_{2}} \right)}} \right),$one can solve for an expression of k₂, $\begin{matrix}{k_{2} = {\exp\left( {A_{2} - \frac{E_{2}}{RT} - {{\frac{\Delta\quad C_{P}}{R}\left\lbrack {\frac{T_{m}}{T} - 1 + {\ln\left( \frac{T}{T_{m}} \right)}} \right\rbrack}.}} \right.}} & (17)\end{matrix}$The above treatment has been well supported by experimentalobservations. Note that it is assumed that the ΔC_(P) here isapproximately equal to the ΔC_(p) measured in the experiment with ureaand it is the only contribution that modifies Arrhenius rates. The termE₂ is the refolding activation energy when temperature equals T_(m).

The kinetic rate constants, k₃ or k₄, can be determined according toEyring's model and shown to conform to a similar expression of the formfor k₂. Knowing that an equilibrium between the U and D₁* or U and D₂*is proposed to exist, there must be corresponding equilibrium constantsK₂₃ and K₂₄ that describe this part of the system. These constantsassume the same for as K₁₂ but replace the parameters {ΔH_(m), T_(m),ΔC_(P)} with {ΔH_(m23), T_(m23), ΔC_(P23)} for k₃ and {ΔH_(m24),T_(m24), ΔC_(P24)} for k₄. Multiplying the resulting equilibriumconstants in each case by k_(x) ( kBT/2xh) (where the subscript x=3 or 4corresponding to either k₃ or k₄; k_(x) is the transmissionco-efficient) Raising the multiplication factor to an exponent, it canbe combined with the exponential form of eqn (16) to yield,$k_{3} = {\exp\begin{pmatrix}{\left\lbrack {\frac{\Delta\quad H_{m\quad 23}}{{RT}_{m\quad 23}} - {\frac{\Delta\quad C_{P\quad 23}}{R}\left( {1 + {\ln\quad T_{m\quad 23}}} \right)} + {\ln\quad\frac{k_{3}k_{B}}{2{\pi\hslash}}}} \right\rbrack -} \\{\frac{{\Delta\quad H_{m\quad 23}} - {\Delta\quad C_{P\quad 23}T_{m\quad 23}}}{RT} +} \\{\frac{{\Delta\quad C_{P\quad 23}} + R}{R}\ln\quad T}\end{pmatrix}}$ $k_{4} = {\exp\begin{pmatrix}{\left\lbrack {\frac{\Delta\quad H_{m\quad 24}}{{RT}_{m\quad 24}} - {\frac{\Delta\quad C_{P\quad 24}}{R}\left( {1 + {\ln\quad T_{m\quad 24}}} \right)} + {\ln\quad\frac{k_{4}k_{B}}{2{\pi\hslash}}}} \right\rbrack -} \\{\frac{{\Delta\quad H_{m\quad 24}} - {\Delta\quad C_{P\quad 24}T_{m\quad 24}}}{RT} +} \\{\frac{{\Delta\quad C_{P\quad 24}} + R}{R}\ln\quad T}\end{pmatrix}}$The resulting k3 and k4 expressions are grouped to show two temperaturedependent terms and one temperature independent constant term. Thetemperature independent or constant term is enclosed within the bracketsseparated from the two temperature dependent terms in the exponential.Among the temperature dependent terms, one is inversely proportional totemperature and the other is proportional to 1n.T. Among the parameters,ΔH_(m2x), T_(m2x, ΔCP) _(2x, kx), we have adopted the following definedrelations that transform the expressions above into equations for k₃ andk₄ that are similar to eqn (17). For k₃, we defineΔ  C_(P)^(D₁) ≡ Δ  C_(P  23) + R, E₃ − Δ  C_(P)^(D₁)T_(m) ≡ Δ  H_(m  23) − Δ  C_(P  23)T_(m  23)  and$A_{3} \equiv \left\lbrack {\frac{\Delta\quad H_{m\quad 23}}{{RT}_{m\quad 23}} - {\frac{\Delta\quad C_{P\quad 23}}{R}\left( {1 + {\ln\quad T_{m\quad 23}}} \right)} + {\frac{\Delta\quad C_{P}^{D_{1}}}{R}\left( {1 + {\ln\quad T_{m}}} \right)} + {\ln\quad\frac{\kappa_{3}k_{B}}{2{\pi\hslash}}}} \right\rbrack$Similar relations for k₄ can be used to derive the final equations forthe rate constants shown below, $\begin{matrix}{k_{3} = {\exp\left( {A_{3} - \frac{E_{3}}{RT} + {\frac{\Delta\quad C_{P}^{D_{1}}}{R}\left\lbrack {\frac{T_{m}}{T} - 1 + {\ln\left( \frac{T}{T_{m}} \right)}} \right\rbrack}} \right.}} & (18) \\{k_{4} = {\exp\left( {A_{4} - \frac{E_{4}}{RT} + {\frac{\Delta\quad C_{P}^{D_{2}}}{R}\left\lbrack {\frac{T_{m}}{T} - 1 + {\ln\left( \frac{T}{T_{m}} \right)}} \right\rbrack}} \right.}} & (19)\end{matrix}$Thus far, we have expressed the kinetic equations for the system interms of a set of 11 independent parameters: {A₁, E₁, A₂, E₂, A₃, E₃,A4, E₄, ΔC_(P), ΔC_(P) ^(D) ¹ , ΔCD_(P) ^(D) ² }. As will be shown inthe next section, the excess heat capacity can also be expressed interms of the same set of parameters. From these parameters T_(m) and ΔHmcan be calculated.

3. The Observables: C_(P) and A_(gg)

Having described the kinetic model of the system, it was important tofind expressions for the excess molar heat capacity C_(P) and the totalmass of the aggregates during the reaction in terms of these kineticparameters.

In the Time-Temperature Aggregation experiment, the total mass of theaggregates is proportional to 1−N(t)−U(t) or D. Here aggregation isdefined by the expression,A _(gg)(T, t)=D.  (20)

It can be calculated by solving the differential eqns (6)-(8). From theaggregation equation (eqn (20)), one can derive an initial rate ofaggregation as $\begin{matrix}{{R_{0}(T)} = \left. \frac{\mathbb{d}\left( {{{Agg}\left( {T,t} \right)}\left\lbrack N_{0} \right\rbrack} \right.}{\mathbb{d}t} \middle| {}_{t = 0}{\approx \left( {{{{Agg}\left( {T,t_{1}} \right)}/t_{1}}*\left\lbrack N_{0} \right\rbrack} \right.} \right.} & (21)\end{matrix}$where t₁ is the first time point in the measurement for a giventemperature. Agg(T,0) is assumed to be zero and [N₀] is the initialmolar concentration of the protein.

In the DSC experiment, the measured quantity is the molar excess heatcapacity under constant pressure.$C_{P} = {\left( \frac{\partial h}{\partial T} \right)P}$where h is the total molar excess enthalpy of the system involved in thedenaturation process. In the proposed model, the states involved in theendotherm are N, U, D₁, and D₂ with N=U=D₁+D₂=1. $\begin{matrix}\begin{matrix}{h = {{h_{N}N} + {h_{U}U} + {h_{D_{1}}D_{1}} + {h_{D_{2}}D_{2}} - h_{N}}} \\{= {{h_{N}N} + {h_{U}U} + {h_{D_{1}}D_{1}} + {h_{D_{2}}D_{2}} - {h_{N}\left( {N + U + D_{1} + D_{2}} \right)}}} \\{= {{\left( {h_{U} - h_{N}} \right)U} + {\left( {h_{D_{1}} - h_{N}} \right)D_{1}} + {\left( {h_{D_{2}} - h_{N}} \right)D_{2}}}}\end{matrix} & (22)\end{matrix}$where h_(x), (x=N, U, D₁, D₂) are the enthalpies of the correspondingensemble states, consisting of native, unfolded, and the final aggregatestates of the first and second order components of the reaction.

Now one can write the observed C_(P) more explicitly as $\begin{matrix}{{C_{P}\left( {v,T} \right)} = {{\left( {h_{U} - h_{N}} \right)\frac{\partial D_{1}}{\partial T}} + {\left( {h_{D_{1}} - h_{N}} \right)\frac{\partial D_{2}}{\partial T}} + {\frac{\partial\left( {h_{U} - h_{N}} \right)}{\partial T}U} + {\frac{\partial\left( {h_{D_{1}} - h_{N}} \right)}{\partial T}D_{1}} + {\frac{\partial\left( {h_{D_{2}} - h_{N}} \right)}{\partial T}D_{2}}}} & (23)\end{matrix}$Since the temperature is increased linearly in time for each scan rateν, i.e. T=T₀+νt, the temperature derivatives can be expressed in termsof the time derivatives and the scan rate using the chain rule, e.g.$\frac{\partial N}{\partial T} = {{\frac{\partial N}{\partial t}\frac{\partial t}{\partial T}} = {\frac{1}{v}\overset{.}{N}}}$The same applies to U, D₁, and D₂. Using eqns (6)-(8), the equation canbe written as $\begin{matrix}{{C_{P}\left( {v,T} \right)} = {{\left( {h_{U} - h_{N}} \right)\left( {{- \frac{1}{v}}\overset{.}{N}} \right)} + {\left( {h_{D_{1}} - h_{U}} \right)\frac{k_{3}}{v}U} + {\left( {h_{D_{2}} - h_{U}} \right)\frac{k_{4}}{v}U^{\quad 2}} + {\frac{\partial\left( {h_{U} - h_{N}} \right)}{\partial T}U} + {\frac{\partial\left( {h_{D_{1}} - h_{N}} \right)}{\partial T}D_{1}} + {\frac{\partial\left( {h_{D_{2}} - h_{N}} \right)}{\partial T}D_{2}}}} & (24)\end{matrix}$

The native state is the standard state of the protein at standardtemperature and pressure and an energy reference point. (h_(U)−h_(N))can be expressed in terms of the unfolding enthalpy and ΔC_(P) (see FIG.2).(h _(U) −h _(N))=ΔH _(m) +ΔC _(P)(T−T _(m))=E₁ −E ₂ +ΔC _(P)(T−T_(m)),  (25)it then follows that $\begin{matrix}{\frac{\partial\left( {h_{U} - h_{N}} \right)}{\partial T} = {\Delta\quad C_{p}}} & (26)\end{matrix}$Similarly, with the assumption that no entholpy change takes place in D*to D transitions (h_(D) ₁ ,−h_(U)) and (h_(D2) ₁ ,−h_(U)) can bedetermined by the equilibrium constants K₂₃ and K₂₄ defined in theprevious section. Using the van't Hoff equation, $\begin{matrix}\begin{matrix}{\left( {h_{D_{1}} - h_{U}} \right) = {{- R}\frac{{\partial\ln}\quad K_{23}}{{\partial 1}/T}}} \\{= {{\Delta\quad H_{m\quad 23}} + {\Delta\quad{C_{P\quad 23}\left( {T - T_{m\quad 23}} \right)}}}} \\{\approx {E_{3} + {\Delta\quad{C_{P}^{D_{1}}\left( {T - T_{m}} \right)}}}}\end{matrix} & (27) \\\begin{matrix}{\left( {h_{D_{2}} - h_{U}} \right) = {{- R}\frac{{\partial\ln}\quad K_{24}}{{\partial 1}/T}}} \\{= {{\Delta\quad H_{m\quad 24}} + {\Delta\quad{C_{P\quad 24}\left( {T - T_{m\quad 24}} \right)}}}} \\{\approx {E_{4} + {\Delta\quad{C_{P}^{D_{2}}\left( {T - T_{m}} \right)}}}}\end{matrix} & (28)\end{matrix}$consequently, $\begin{matrix}\begin{matrix}{\frac{\partial\left( {h_{D_{1}} - h_{N}} \right)}{\partial T} = \frac{\partial\left\lbrack {\left( {h_{D_{1}} - h_{U}} \right) + \left( {h_{U} - h_{N}} \right)} \right\rbrack}{\partial T}} \\{= {{\Delta\quad C_{P}^{D_{1}}} + {\Delta\quad C_{P}}}}\end{matrix} & (29) \\\begin{matrix}{\frac{\partial\left( {h_{D_{2}} - h_{N}} \right)}{\partial T} = \frac{\partial\left\lbrack {\left( {h_{D_{2}} - h_{U}} \right) + \left( {h_{U} - h_{N}} \right)} \right\rbrack}{\partial T}} \\{= {{\Delta\quad C_{P}^{D_{2}}} + {\Delta\quad{C_{P}.}}}}\end{matrix} & (30)\end{matrix}$

Substituting eqn (25-30 into eqn (24), we have an expression for C_(p)in terms of scan rate and temperature, $\begin{matrix}{{C_{P}\left( {v,T} \right)} = {{\left( {{\Delta\quad H_{m}} + {\Delta\quad{C_{P}\left( {T - T_{m}} \right)}}} \right)\left( {{- \frac{1}{v}}\overset{.}{N}} \right)} + {\Delta\quad C_{P}U} + {\frac{k_{3}}{v}\left( {E_{3} + {\Delta\quad{C_{P}^{D_{1}}\left( {T - T_{m}} \right)}}} \right)U} + {\frac{k_{4}}{v}\left( {E_{4} + {C_{P}^{D_{2}}\left( {T - T_{m}} \right)}} \right)U^{2}} + {\left( {{\Delta\quad C_{P}^{D_{1}}} + {\Delta\quad C_{P}}} \right)D_{1}} + {\left( {{\Delta\quad C_{P}^{D_{2}}} + {\Delta\quad C_{P}}} \right)D_{2}}}} & (31)\end{matrix}$

The first line of eqn (31) contains the familiar terms describing the2-state reversible system. The terms on the second line of eqn (31) arenecessary to describe the contribution of the aggregation step to theoverall heat absorption. When k₃ and k₄ are zero, so are D₁ and D₂, theequation reduces to the fully reversible case. When the sum, ΔC_(P) ^(D)¹ +ΔC_(P) or ΔC_(P) ^(D) ² +ΔC_(P) do not equal zero, the last two termsgive a non-vanishing baseline shift that can contribute to a negativenet influence on the transition baseline. The negative influence on thetransition baseline can occur when ΔC_(P) ^(D) ¹ or ΔC_(P) ^(D) ² resultfrom buried hydrophobic surface as would be expected in the aggregationprocess. After subtraction of the transition baseline, the last twoterms in eqn (31) are of no consequence. Therefore, the fitted datashown in FIG. 3 are based on this equation with the last two termsexcluded.

III. Applications

Both methods and computer apparatus for implementing the new approachwill be described below.

A. Example Method

An example of the basic steps of providing measurements ofconformational change, using those measurements to rigorously deriveuseful parameters, and using the parameters is set forth in FIG. 8, andis described below.

1. Providing Measurements of Conformational Change

The methods of data acquisition will be discussed after a description ofsome of the various materials in connection with which the method can beused.

a. Compositions

The methods of the invention can be used with any biologically activematerial that is capable of a conformational change and, in certainembodiments, unfolding, due to a thermal change and that exhibitsmulti-state behavior provided that such behavior can be measured by asuitable biophysical procedure as described further below. In certainembodiments wherein the material exhibits behavior that has more thantwo states, the behavior is dissected into two-state segments.

The biologically active material may be a polypeptide or protein. Suchmaterials include but are not limited to hormones, cytokines,hematopoietic factors, growth factors, antibodies, antiobesity factors,trophic factors, anti-inflammatory factors, and enzymes (see also U.S.Pat. No. 4,695,463 for additional examples of useful biologically activeagents).

The material could also include antibodies, antibody-like molecules,interferons (see, U.S. Pat. Nos. 5,372,808, 5,541,293 4,897,471, and4,695,623 hereby incorporated by reference including drawings),interleukins (see, U.S. Pat. No. 5,075,222, hereby incorporated byreference including drawings), erythropoietins (see, U.S. Pat. Nos.4,703,008, 5,441,868, 5,618,698 5,547,933, and 5,621,080 herebyincorporated by reference including drawings), granulocyte-colonystimulating factors (see, U.S. Pat. Nos. 4,810,643, 4,999,291,5,581,476, 5,582,823, and PCT Publication No. 94/17185, herebyincorporated by reference including drawings), stem cell factor (PCTPublication Nos. 91/05795, 92/17505 and 95/17206, hereby incorporated byreference including drawings), and leptin (OB protein) (see PCTpublication Nos. 96/40912, 96/05309, 97/00128, 97/01010 and 97/06816hereby incorporated by reference including figures).

The materials might also include substances like insulin, gastrin,prolactin, adrenocorticotropic hormone (ACTH), thyroid stimulatinghormone (TSH), luteinizing hormone (LH), follicle stimulating hormone(FSH), human chorionic gonadotropin (HCG), motilin, interferons (alpha,beta, gamma), interleukins (IL-1 to IL-12), tumor necrosis factor (TNF),tumor necrosis factor-binding protein (TNF-bp), brain derivedneurotrophic factor (BDNF), glial derived neurotrophic factor (GDNF),neurotrophic factor 3 (NT3), fibroblast growth factors (FGF),neurotrophic growth factor (NGF), bone growth factors such asosteoprotegerin (OPG), insulin-like growth factors (IGFs), macrophagecolony stimulating factor (M-CSF), granulocyte macrophage colonystimulating factor (GM-CSF), megakaryocyte derived growth factor (MGDF),keratinocyte growth factor (KGF), thrombopoietin, platelet-derivedgrowth factor (PGDF), colony simulating growth factors (CSFs), bonemorphogenetic protein (BMP), superoxide dismutase (SOD), tissueplasminogen activator (TPA), urokinase, streptokinase, and kallikrein.

In certain embodiments, the biologically active material will beformulated as a pharmaceutical composition comprising effective amountsof the biologically active material, or derivative products, togetherwith pharmaceutically acceptable excipients, for example, diluents,preservatives, solubilizers, emulsifiers, adjuvants, and/or carriersneeded for administration. The optimal pharmaceutical formulation for adesired biologically active material can be determined by one skilled inthe art depending upon the route of administration and desired dosage.See, e.g., Remington's Pharmaceutical Sciences (Mack Publishing Co.,18th Ed., Easton, Pa., pgs. 1435-1712 (1990)).

The material could also be a solution that could be in either liquid orsolid amorphous state, or in a glassy state. Sometimes, the material mayhave been lyophilised.

b. Data Acquisition

Conformational change (e.g., unfolding) of these compositions can bemeasured as a function of temperature varied with time. In some case, itmay be important for temperature to be varied uniformly with time.

A variety of different thermodynamic parameters can be determined,including: enthalpy of the transition (ΔH) or the free energy (ΔG);ΔC_(p) (the change in heat capacity between native and denatured statesof the material); and the temperature at which about 50% of the moleculeis unfolded and about 50% of the molecule is in its native state (forexample, T_(m), T_(g), or C_(max)).

When the biologically active material is a nucleic acid, the meltingtemperature of the nucleic acid can be measured. In other cases, thehybridization properties of the nucleic acid can be analyzed.

Differential scanning calorimetry or DSC can might be used to determinethe temperature of denaturation if the system is fully reversible andthe scan rate does not exceed the rate of unfolding. See, Privalov, P.L.; Khechinashvili, N. N. J. Mol. Biol. 1974, 86, 665-684 and Lepock, J.R.; Ritchie, K. P.; Kolios,.M. C.; Rodahl, M; Heinz K. A.; Kruuv, J.Biochemistry 1992, 31, 12706-12712. In other embodiments, circulardichroism or CD can be used to analyze the composition. The measuredquantity in the CD experiment is molar ellipticity.

Fluorescence can sometimes be used to analyze the composition.Fluorescence encompasses the release of energy in the form of light orheat, the absorption of energy in the form or light or heat, changes inturbidity, and changes in the polar properties of light. Specifically,the term refers to fluorescent emission, fluorescent energy transfer,absorption of ultraviolet or visible light, changes in the polarizationproperties of light, changes in the polarization properties offluorescent emission, changes in turbidity, and changes in enzymeactivity. Fluorescence emission can be intrinsic to a protein or can bedue to a fluorescence reporter molecule. For a nucleic acid,fluorescence can be due to ethidium bromide, which is an intercalatingagent. Alternatively, the nucleic acid can be labeled with afluorophore.

HPLC methods can also sometimes be used to analyze the conformationalchange of the biologically active material.

In this example, we began with the DSC experiment (block 10 in FIG. 8),which provided measured data for C_(P) (T,ν) (block 12). Beforeproceeding, we evaluated whether we had enough information for making agood initial estimate of the initial parameters (14). In this case, wewanted both to use C_(P) (T,ν) and A_(gg) (T,t) for estimating theinitial parameters. Accordingly, we obtained further data from othersources. We used an SEC time aggregation study (16) to provideaggregation data (18), giving us data on both C_(P) (T,ν) and A_(gg)(T,t) (block 20).

The rate of aggregation was derived from a plot showing the amount ofaggregate measured by SEC as a function of time as depicted in FIG. 4using eqn (21). At low temperatures between 30 and 50° C., linear ratesof reaction were easily ascertained since there was no deviation fromlinearity during the period of time examined. However, attemperatures >58° C., the initial reaction rate was calculated accordingto eqn (20). The observed rate constants obtained from these data wereconverted to units of M/sec and then subsequently evaluated in the formof an Arrhenius plot (FIG. 5A).

It has been shown that it is possible to fit the data with two lines,reflecting approximate Arrhenius behaviour above and below the apparentmelting temperature (as depicted in FIG. 5A). The data points thatdetermine the lines were found by optimizing both correlationcoefficients. From the slope of the lines describing these two regions,one is able to characterize the aggregation reaction kinetics with twoactivation energies. At low temperatures (below T_(m)), an activationenergy of 100 kcal/mol was obtained and at high temperatures (aboveT_(m)), the activation energy was found to be approximately 28 kcal/mol.

The Arrhenius data in FIG. 5A clearly show a break from linearity in thereaction rate around the melting temperature, indicating two differentactivation energies at two different temperature regions, separatedapproximately by T_(m). This behavior can be explained by a rateapproximation from the kinetic model (eqn (6)-(8)). When k₃<<k₂ andk₄<<k₃ (valid at low temperatures below the T_(m)), the equilibriumbetween N and U predominates as expected and the aggregation can bedescribed the an effective rate coefficient$k^{\prime} = {\frac{k_{1}k_{3}}{k_{1} + k_{2} + k_{3}}.}$The effective rate is plotted as the dashed line in FIG. 5. It can beseen that this approximation is very good at low temperature when(T<<T_(m), k₂/k₁>>1 therefore k′≈k₃k₁/k₂). When T >>T_(m), k′≈k₃,although the rate approximation for aggregates is not as good in thistemperature region, one can still obtain a qualitative estimate of k₃from the experimental data. It should be noted that our numericalfitting was performed over the entire time-sequence of study and notjust limited to the initial rates. The initial rates are used toestimate the starting parameters for the nonlinear fitting and used forchecking the consistency of the final results.

In the DSC experiment, the apparent melting temperature T_(app) dependson the scan rate of the irreversible system. The scan-rate dependentactivation energy of the T_(app) was also examined to ascertain therelevance to the activation energy obtained from the Arrhenius plot ofthe time-temperature studies. The results obtained from a plot of In└ν/T_(app) ²┘ as a function of 1/T as proposed by Sánchez et.al. yieldeda straight line with an associated activation energy of about 89kcal/mol. This treatment appears to yield an activation energy that isconsistent with the low temperature fit in the SEC experiment. Namely, arespectable result that compares well with the activation energyobtained from the time-temperature studies.

For illustration purposes, the scan-rate method of Sánchez-Ruiz andcoworkers is plotted as solid circles in FIG. 5B [within the region from0.0029 (61.3° C.) to 0030 (56.9° C.)]. The lower activation energy ofthe scan-rate dependent result may exhibit some bias imposed by theregion of curvature so that extrapolation to commercially important lowtemperatures (i.e., 2-8° C.) would result in overestimates ofaggregation reaction rates.

2. Obtaining Parameters

As discussed below, useful parameters can be rigorously obtained bymaking an initial estimate, fitting the data to obtain fittedparameters, checking the uniqueness of those parameters, and thenassessing their variability.

a. Making Initial Estimates

The measured A_(gg) (T,t) data is shown in FIG. 1. Together, C_(P) (T,v)and A_(gg) (T,t) enabled one of skill in the art to make an initialrough estimate of the parameters of interest (blocks 22, 23 in FIG. 8).Those initial rough estimates were used as the starting point forfitting the data.

b. Fitting the Data

Exemplary parameter estimation methods that can be used to comparepredicted measurements of conformational change to actual measurementsof conformational change include non-linear least squares fittingmethod, maximum likelihood method, non-linear least norms method, andother methods known in the art.

In this example, eqn (31) was used to fit the experimental data toextract all the Arrhenius parameters. In general, the originaldifferential equations were solved with initial parameters and initialconditions (block 26 in FIG. 8), giving N, U, and D for the experiments(28). From these, theoretical values of Cp(T,v) and Agg(T,t) werecalculated (30, 32). These calculated values were compared to themeasured data, and the sum of the squares of the differences werecalculated (34). The sum was evaluated to determine if it was minimal(36). When it was not minimal, a new set of best guess parameters wascreated (38, 40) and a new iteration began, the process continuing untilthe sum of squares was minimal.

In this case, a nonlinear least square fitting routine (Isqnonlin inMatlab) was applied to minimize a linear combination of the twosum-of-squares functions simultaneously. The first represents thedifference between the calculated values and the observed values in theDSC experiment, and the second represents the difference between thecalculated and observed values in the SEC experiment. Denoting thecalculated values with a superscript calc and the observed values with asuperscript exp, the two sum of squares functions can be written as,$\begin{matrix}{\chi_{1} = {\sum\limits_{i = 1}^{4}{\sum\limits_{i = 1}^{n_{i}}{w_{i}{{{C_{P}^{calc}\left( {v_{i},T_{j}} \right)} - {C_{P}^{\exp}\left( {v_{i},T_{j}} \right)}}}^{2}}}}} & (32)\end{matrix}$where ν_(i), (i=1, 2, 3, 4) is the designated scan rate and n_(i), (i=1,2, 3, 4) represents the corresponding data points of each scan. The termw_(i), is an optional weighting factor to ensure each data point iscounted appropriately. The reason for the weighting factor is to accountfor the collection of more data points at the slower scan rate than atthe faster scan rate. Without weighting, each data point contributesequally to the sum-of-squares function. This leads to more contributionsfrom the slower scan data. The weighting factor of w_(i)=1/n_(i) makesthe contribution of each scan equal. $\begin{matrix}{\chi_{2} = {\sum\limits_{i = 1}^{9}{\sum\limits_{j = 1}^{n_{i}}{{{\ln\left( {{Agg}^{calc}\left( {T_{i},t_{j}} \right)} \right)} - {\ln\left( {{Agg}^{\exp}\left( {T_{i},t_{j}} \right)} \right)}}}^{2}}}} & (33)\end{matrix}$where i=1 to 9 represents the 9 temperatures under which the aggregationexperiment took place. We used n_(i)=3, i.e. the first three points ateach temperature to do the fitting.

Note that either x₁ or x₂ can be used alone for fitting the parameters.In fact that is what was often done in the literature. But thesensitivity of the two observed quantities to each parameter isdifferent. In order to best constrain the parameter space, we performeda simultaneous fitting for both. In other words we minimized thefollowing quantity:χ=χ₁ +F·χ ₂  (34)where F is an arbitrary scaling factor selected to make contributionsfrom χ_(i), and F·χ₂ to χ similar in order of magnitude, since the twoquantities have different dimensions and would be otherwiseincomparable.

After performing the procedure of the nonlinear least square fitting, weobtained numerical values for all the parameters in the model system. Asdiscussed above, since the aggregation reaction is entropically driven,the transition from D₁* and D₂* to the final D₁ and D₂ states wasspontaneous, driven by the removal of hydrophobically exposed surfacesduring aggregation. Little or no heat was released ΔH_(agg) ^(‡)≈0) inthis process, but the entropy of the solution was increased. So thechange in Gibbs free energy ΔG_(agg) ^(‡) (see FIG. 2) can be expressedas $\begin{matrix}\begin{matrix}{{\Delta\quad G_{agg}^{\ddagger}} \approx {{- T}\quad\Delta\quad S_{agg}^{\ddagger}}} \\{= {- {T\left( {{\Delta\quad{S_{D_{1}^{*}D_{1}}^{\ddagger} \cdot D_{1}}} + {\Delta\quad{S_{D_{2}^{*}D_{2}}^{\ddagger} \cdot D_{2}}}} \right)}}}\end{matrix} & (35)\end{matrix}$

The relevant parameters are displayed in Table 1. TABLE 1 Parametersobtained from fitting the model to both DSC traces and SEC aggregationcurves simultaneously. $\begin{matrix}A_{1} \\{\ln\left( \frac{1}{\min} \right)}\end{matrix}\quad$ $\begin{matrix}E_{1} \\\frac{kcal}{mol}\end{matrix}\quad$ $\begin{matrix}A_{2} \\{\ln\left( \frac{1}{\min} \right)}\end{matrix}\quad$ $\begin{matrix}E_{2} \\\frac{kcal}{mol}\end{matrix}\quad$ $\begin{matrix}A_{3} \\{\ln\left( \frac{1}{\min} \right)}\end{matrix}\quad$ 115.4 ± 1.7 76.6 ± 1.2 1.0 ± 11.7 2.3 ± 8.0 68.9 ±6.9 $\begin{matrix}E_{3} \\\frac{kcal}{mol}\end{matrix}\quad$ $\begin{matrix}A_{4} \\{\ln\left( \frac{1}{\min} \right)}\end{matrix}\quad$ $\begin{matrix}E_{4} \\\frac{kcal}{mol}\end{matrix}\quad$ $\begin{matrix}{\Delta H}_{m} \\\frac{kcal}{mol}\end{matrix}\quad$ 46.8 ± 4.4 228.1 ± 22.6 151.7 ± 15.0 74.3 ± 6.8$\begin{matrix}T_{m} \\K\end{matrix}\quad$ $\begin{matrix}{\Delta Cp} \\\frac{kcal}{{mol} \cdot K}\end{matrix}\quad$ $\begin{matrix}{\Delta C}_{P}^{D_{1}} \\\frac{kcal}{{mol} \cdot K}\end{matrix}\quad$ $\begin{matrix}{\Delta C}_{P}^{D_{2}} \\\frac{kcal}{{mol} \cdot K}\end{matrix}\quad$ 326.6 ± 1.4 1.3 ± 0.7 −0.2 ± 0.5 −5.8 ± 1.6

The rate constant profiles as a function of temperature are shown inFIG. 6 with corresponding activation energies in Table 1. Consideringconditions below T_(m) in FIG. 6, the model depicts k₄ (with associatedactivation energy, E₄) as least important based on a slower reactionrate and correspondingly high-energy barrier (152 kcal/mol). E₃ on theother hand with associated rate constant k₃ has a lower activationenergy (47 kcal/mol) than either E₁ or E₄, and therefore predominatesthe aggregation pathway, rate limited only by the supply of aggregationcompetent spaces D₁ and unfolded forms of the protein. This is explicitfrom the fact that k₁ is rate limiting and slower than k₃. Hence attemperatures below the T_(m), solution conditions that tend to stabilizethe native state will greatly impede formation of aggregate by either E₃or E₄ routes of aggregation.

From the post-T_(m) data fit of FIG. 5A, an activation energy of 28kcal/mol was obtained. This activation energy was ascribed to a changein aggregation rate that depended upon unfolding. Although tempting toassert this 28 kcal/mol activation energy to E₃ as described by theLepock model, we have found that the theory contained in our modelpresents a more complex picture of the transition states post-unfolding.As temperature exceeds the T_(m), k₄ becomes dominant overtaking k₃ eventhough the activation energy barrier is greater than E₃. It is importantto realize that the U state is sufficiently populated to supply both D₁*and D₂* at or above the T_(m). The reaction pathway driven by k₃ and k₄are no longer rate limited by k₁. Hence the rate, k₃, becomes lessprominent, while k₄ becomes the rate determining factor post-T_(m) asdescribed in FIG. 6 even though its activation energy is greater thank₃. The result leads to a reasonable fit of the data as depicted in FIG.5B.

It can be seen from FIG. 3 that there is good agreement between eqn (31)and the experimental data. It can be seen that they span a range of 10orders of magnitude. For that reason, a system of stiff differentialequations (eqns 6-8) needs to be solved. Matlab ODE Solver ode15s wasused. The temperature where k₁ and k₂ cross corresponds to the T_(m)(˜53.5° C.) for a fully reversible unfolding system that involves onlythe native and unfolded states.

The comparison of calculated aggregation based on the model and theexperimentally obtained SEC data is shown in FIG. 4 and FIG. 5B. Notethe temperature associated with each data point has a variability of ±1°C. as shown in the plot. Regarding the calculated rates shown in FIG. 4,the curves above T_(m) are calculated from an initial conditionN(t=0)=1, U(t=0)=D₁(t=0)=D₂(t=0)=0 assuming a linear heating time of 60sec from T₀=25° C., while the curves below T_(m) were calculated with anequilibrium between N and U at t=0, i.e. N(t=0)/U((t=0)=k₂/k₁. TABLE 2Comparison of initial rate predicted at 34° C. for the 1 mg/mL solutionby differing methods. For pre-T_(m) fit, see FIG. 5A. The rate for thismodel is calculated by converting A_(gg) (T, t₁)/t₁ * [N₀] to thecorrect units. Experiment Sánchez-Ruiz Pre-T_(m) fit eqn (21)rate(M/sec) 8.5 · 10⁻¹³ 3.8 · 10⁻¹² 6.3 · 10⁻¹³ 6.2 · 10⁻¹³

A comparison of the initial rate obtained from the Time-Temperatureexperiment at 34° C. to the rates obtained by three different methods islisted in Table 2. The aggregation rate predicted by the model (eqn(21)) and the pre-T_(m) fit best represents the experimentallydetermined aggregation rate in comparison to the Sánchez-Ruiz model.

There is another check for the consistency of all our parameters: thetotal heat (enthalpy) of the reaction. The experimental measurement canbe directly calculated by: $\begin{matrix}{{\Delta\quad H} = {\int_{T_{0}}^{T_{F}}{C_{P}\quad{\mathbb{d}T}}}} & (36)\end{matrix}$where T₀ and T_(F) are the beginning and final temperatures of thetransition envelope. From eqn (31), we can obtain approximateexpressions for this quantity:ΔH≈(E ₁ −E ₂)+E ₃ ·D ₁+2E ₄ ·D ₂  (37)

where E₁−E₂=ΔH_(m), the unfolding reaction enthalpy. D₁ and D₂ areobtained at the end of our numerical integration of eqn (6)-(8). Thecomparison is shown in Table 3. It can be seen that there is goodagreement in ΔH between the experimental and calculated results. TABLE 3Comparison of total enthalpy measured (top row), approximation from eqn(37) (middle row) and the simulated values from eqn (31) (bottom row). υ(degree/min) 0.25 0.5 1.0 1.5 eqn (36) 128 129 138 130 kcal/mol eqn (37)123 131 141 147 kcal/mol eqn (31) 125 131 138 140 kcal/mol

All the experimental data discussed so far were obtained with a fixedconcentration of 2 mg /mL. Interestingly, the model can be applied todescribe the concentration dependence of the aggregation rate (see eqns(12)-(13)). FIG. 7 shows the predicted and the experimentally determinedconcentration response at two different temperatures corresponding todifferent time durations. It shows the simple assumption that k₃ dependslinearly on the concentration works fairly well at the lowerconcentration range but tends to overestimate aggregation rates athigher concentrations. As for why it deviates from the experimentalresults more significantly at the highest concentrations tested isunclear. A possible explanation may be that at higher concentrationsthere is a greater tendency for the protein to self-associate in thesolution phase. Such molecular crowding as the protein concentration isincreased can lead to an augmentation of aggregation. However, a portionof non-covalent self-associated aggregates can be reversible entities insolution that are not picked up by the SEC method and therefore may beobserved lower than what actually exists in solution. This could occuras a result of dilution or during passage through the size exclusioncolumn. The disparity between aggregation in solution and thatdetermined by SEC would then be expected to increase with concentrationwhere the theoretical prediction exceeds SEC aggregation results.

c. Checking the Uniqueness

We paid special attention to the question of identifiability: theability to guarantee that all eleven parametersP={A₁, E₁, A₂, E₂, A₃, E₃, A_(4, E) ₄, ΔC_(P),ΔC_(P) ^(D) ¹ , ΔC_(p)^(D) ² }are uniquely determined by minimizing eqn (34). (If the parameters werenonidentifiable, then different values for the eleven parameters couldproduce the same sum of squares.) (Consequently, T_(m) and ΔHm are alsoidentifiable with reasonable variances.) It is not possible to guaranteethis globally, but we can guarantee it locally near the calculatedparameters p₀ yielding the minimum of χ (eqn (34)).

One possible way to assess model identifiability is laid out in FIG. 9.In that routine, there is first a symbolic calculation of the system ofvariational differential equations (block 52). Then there is a numericalsolution of the variational differential equations at the values of thefitted parameters (54). Then a linearization of the mapping of theparameters to the observables at the values of the fitted parameters isbuilt (56). This can be done, for example, by solving variationaldifferential equations. Then a decision is made if the linearization isnondegenerate (58). If it is, then the system is identifiable (60). Ifnot, then the width of the preimage of the unit cube in the functionalspace is checked to see if it is not infinity (62). If so, the system isidentifiable (60). If not, the system is not identifiable (64).

We used variational methods to calculate the partial derivatives$\frac{\partial{C_{P}\left( T_{i} \right)}}{\partial_{Pj}}$for each of the 11 parameters j=1, . . . , 11 at each of the ntemperature points T_(i), i=1, . . ., n evaluated by the symbolic andnumerical differential equation solver (in our calculations n was on theorder of 200). This yielded a matrix of size 11 by n. Numericalcalculation showed that the rank of this matrix was maximal size of 11,assuring the nondegeneracy of the defining equations and theidentifiability of the parameters, so the values we calculated areindeed uniquely determined.

The numerical calculations showing that the rank of the matrix ismaximal possible can be produced in different ways. For example it canbe done using norms in the space of parameters and in the space ofobservables. The norm is a function that relates to each vector in thespace a length of the vector.

One can introduce a norm (denote it by NO) in the space of theobservables and a norm (denote it by NP) in the space of the parameters.Consider the subset of all vectors with the length equal to 1. Thissubset is called the unit subset. Denote the unit subset in the space ofthe observables by B1. The unit subset is given by the formulaB1={o:NO(o)=1}. Consider the preimage of the unit subset B1 in the spaceof observables under the mapping ∂pC. Denote this preimage by prmg(B1).The preimage is given by formula prmg(B1)={p: NO(∂pC(p))=1}. The size ofprmg(B1) is defined as the minimal number r such that the subset{p:NP(p)<=r} contains prmg(B1). Denote the size by R. Matrix ∂pC hasmaximal rank if and only if R is less than infinity. In the case when NOand NP are quadratic forms, R is equal to 1/svd where svd is the minimalsingular value in singular values decomposition of the matrix ∂pC. It isalso useful to use other norms NO and NP that give more preciseevaluation of the invertibility of the matrix ∂pC because svd could be(and in fact is) very small for a particular case. For example, the normof maximum of values of the components of parameters and maximum valuesof the components of the observables has been used. The size of theprmg(B1) for these norms is called the width of the prmg(B1).

d. Assessing Variability

Parameter variability (block 30 in FIG. 8) an be analyzed by the routineset out in FIG. 10. As shown there, we first determined the subspace ofthe functions to which errors belong, then projected the errors on thespace of the observables (block 30 in FIG. 10), and found the 95%confidence area (the area in the projected set to which 95% of errorsbelong). We then found a preimage of the 95% confidence set using thelinearization of the mapping from the parameters to the observables (32in FIG. 10), and found widths of the preimage (34 in FIG. 10). Thosewidths established the 95% confidence interval.

3. Using the Parameters

A major hurdle to overcome is the ability to make reasonable estimatesof aggregation half-life at conditions of low temperature storage.Storage conditions of marketed liquid biopharmaceuticals normallyrequire refrigerated temperatures.

Most of the approaches used in this context have relied upon empiricalmodeling methods. Although these methods have applied Arrhenius modelsto make predictions about shelf-life, they have ignored thethermodynamic properties of reactions that can often result innon-Arrhenius behavior. In the study presented, a mechanistic model hasbeen proposed to predict properties of shelf-life as it pertains toaggregation. The commercial viability of a drug product presented tohealth care personnel generally requires a minimum shelf life ofeighteen months when stored at room temperature (20-25° C.) or underrefrigeration (2-8° C.). The shelf life is typically defined as thestorage time after manufacture of the drug product during which the drugexperiences no more than about 5 to 10% degradation, i.e., the drugretains its biological activity.

This model has been applied to better evaluate non-Arrhenius reactionrates of rhuIL-IR(II) aggregation mediated by unfolding. In this firststep taken, the validity of the simulation in regard to appropriatelypredicting the influence of concentration factors, deriving respectablethermodynamic parameters from a partially irreversible process, andemphasizing protein unfolding as a prerequisite to aggregation hassuccessfully explained the aggregation kinetics of the system.

The thermal unfolding enthalpy in the absence of urea exhibited moreheat (˜48 kcal/mol) than could be accounted for in the reversible case(with urea). We have considered other alternative modeling schemes toexplain these results like different unfolded or U states where theenthalpy of urea was lower than the enthalpy in its absence (as dictatedby the experimental results). Although one could still obtain a fit, itwas not as good as the proposed model and there were other issues ofconflict. For example in the case of altered unfolded states, ΔH≠E₁−E₂,and the expectation that it should be a valid equality has beensuggested by Lepock and coworkers. Additionally, when k₃ and k₄ weremade equal to zero (as in the fully reversible case), the theoreticalT_(m) was ˜65° C. (above the highest scan T_(m)), a condition that doesnot satisfy and is far removed from the fully reversible case in urea(˜52.6° C.). Furthermore, the heat gained does coincide with thepopulation of states≧T_(m) where massive aggregation has been confirmedin both the “Time-Temperature” SEC studies as well as those studiesconducted in the calorimeter (in addition to the presence of adeconvoluted peak at ˜60° C. under the unfolding transition on the hightemperature side that is close to the 48 kcal/mol enthalpy increase). Incontrast, by adopting the theory as presented here, there was foundgreater harmony among the experimental observations (i.e., better fit ofthe data), where the 48 kcal/mol increase in enthalpy could be assignedto a deconvoluted peak on the high temperature side representingcontributions from the D states (D₁ and D₂). Moreover, ΔH_(m) moreappropriately agreed with the E₁-E₂ equality (˜74.3 kcal/mol instead of˜54 kcal/mol; closer to the 85 kcal/mol in the urea case). When k₃ andk₄ were set to zero, the T_(m) was approximately ˜54° C., much closer tothe value in urea, but slightly higher as would be anticipated whenchemical denaturant is absent. We examined the case where only a 2^(nd)order aggregation process was simulated and found an unfavorable ΔH_(m)of ˜52 kcal/mol and a predicted T_(m) of 51° C. These values are notvalid since they were not consistent with the measured enthalpy andT_(m) values in the DSC experiment using urea.

The corresponding transition states (D₁* and D₂*) of the D state areinferred from the observation of a noticeable change in kinetics abovethe T_(m). This temperature zone coincides with more rapid aggregationkinetics (massive) than what was observed at temperatures below theT_(m). It testifies to the validity of a constant supply of unfoldedprotein that can rapidly interact by either a 1st or 2nd ordermechanism. Furthermore, justification for E₄ is found by the quality offit on the high temperature side of the DSC endotherm.

The relationship between activation energy and total enthalpy of theunfolding transition described by Sánchez-Ruiz et al and Lepock et alwas found inadequate as an appropriate description of the DSC behaviorat rhuIL-1R (II). Moreover, they did not include influences of theΔC_(P) that contribute to non-Arrhenius aggregation responses. Althoughthe ΔC_(P) does not contribute significantly within the temperatureregime in the vicinity of the T_(m), it can pose more influence at lowertemperature. In contrast to these approaches, the present work hasderived a theoretical treatment that can be applied to DSC data in orderto extract thermodynamically meaningful parameters from a partiallyreversible system. Unlike the work of Sánchez-Ruiz et al., and Lepock etal., this work describes the system in terms of both 1st and 2nd orderreaction properties that depend upon the thermodynamics of unfolding. Ittakes into account the influences of the denaturational heat capacity indescribing the non-Arrhenius kinetics of aggregation that can occur atlow as well as high temperatures. Finally, it satisfactorily describesthe enthalpy and activation energies along the aggregation reactionpathway and lays the groundwork for predicting shelf-life of complexprotein aggregation systems.

The procedure described in the flow chart produced fitted parameters:{A1, E1, A2, E2, A3, E3, A4,E4, ΔCp, ΔCpD1, ΔCpD2}±uncertainties.

From these parameters we can calculate the kinetic rate constants at agiven temperature, for example, 2° C., 4° C. and 8° C., using eqn (14),eqn (17), eqn (18) and eqn (19).

Substituting these constants into equations (6)-(8). Assuming theinitial conditions, N(t=0)=1, U(t=0)=0, D1(t=0)=0, D2(t=0)=0, one cansolve the differential equations (6)-(8) to obtain

1. For a given time duration, typically 2 years for liquidbiopharmaceuticals, the amount of aggregation, Agg(T,t)=D(T,t). Anexample prediction for IL-1R(II) in a specified formulation are listedbelow, t = 2 years 2° C. 4° C. 8° C. Aggregation (%) 3.4e⁻⁶ 8.7e⁻⁶5.9e⁻⁵

-   2. Given a specific level of unacceptable aggregation, the time it    takes to reach that level. For example, if 1% aggregation will not    be acceptable for IL-1R(II), the prediction for this time duration    in the same formulation as in example 1, is greater than 10000    years. Note that in this example an acceptable shelf-life would have    been attained (within a two-year duration). This happens to be an    extraordinary optimal formulation for this protein therapeutic.

The new approach might also be used for other purposes. For example, itmight be used to predict the effect of concentration on shelf life of abiologically active material. Or, it might be used to predict the effectof an excipient on shelf life of a biologically active material. To dothis, the parameters could be determined for a composition comprisingthe biologically active material and one or more excipients. Theseparameters might then be extrapolated for use with the recommendedstorage temperature to yield the predicted shelf-life of thecomposition. These parameters might then be compared to predict theeffect of the excipient on the shelf life of said biologically activematerial.

B. Apparatus

FIG. 9 illustrates an example of a suitable computing system environment100 on which a system for the steps of the claimed method and apparatusmay be implemented. The computing system environment 100 is only oneexample of a suitable computing environment and is not intended tosuggest any limitation as to the scope of use or functionality of themethod of apparatus of the claims. Neither should the computingenvironment 100 be interpreted as having any dependency or requirementrelating to any one or combination of components illustrated in theexemplary operating environment 100.

The steps of the claimed method and apparatus are operational withnumerous other general purpose or special purpose computing systemenvironments or configurations. Examples of well known computingsystems, environments, and/or configurations that may be suitable foruse with the methods or apparatus of the claims include, but are notlimited to, personal computers, server computers, hand-held or laptopdevices, multiprocessor Systems, microprocessor-based systems, set topboxes, programmable consumer electronics, network PCs, minicomputers,mainframe computers, distributed computing environments that include anyof the above systems or devices, and the like.

The steps of the claimed method and apparatus may be described in thegeneral context of computer-executable instructions, such as programmodules, being executed by a computer. Generally, program modulesinclude routines, programs, objects, components, data structures, etc.that perform particular tasks or implement particular abstract datatypes. The methods and apparatus may also be practiced in distributedcomputing environments where tasks are performed by remote processingdevices that are linked through a communications network. In adistributed computing environment, program modules may be located inboth local and remote computer storage media including memory storagedevices.

With reference to FIG. 9, an exemplary system for implementing the stepsof the claimed method and apparatus includes a general purpose computingdevice in the form of a computer 110. Components of computer 110 mayinclude, but are not limited to, a processing unit 120, a system memory130, and a system bus 121 that couples various system componentsincluding the system memory to the processing unit 120. The system bus121 may be any of several types of bus structures including a memory busor memory controller, a peripheral bus, and a local bus using any of avariety of bus architectures. By way of example, and not limitation,such architectures include Industry Standard Architecture (ISA) bus,Micro Channel Architecture (MCA) bus, Enhanced ISA (EISA) bus, VideoElectronics Standards Association (VESA) local bus, and PeripheralComponent Interconnect (PCI) bus also known as Mezzanine bus.

Computer 110 typically includes a variety of computer readable media.Computer readable media can be any available media that can be accessedby computer 110 and includes both volatile and nonvolatile media,removable and non-removable media. By way of example, and notlimitation, computer readable media may comprise computer storage mediaand communication media. Computer storage media includes both volatileand nonvolatile, removable and non-removable media implemented in anymethod or technology for storage of information such as computerreadable instructions, data structures, program modules or other data.Computer storage media includes, but is not limited to, RAM, ROM,EEPROM, flash memory or other memory technology, CD-ROM, digitalversatile disks (DVD) or other optical disk storage, magnetic cassettes,magnetic tape, magnetic disk storage, or other magnetic storage devices,or any other medium which can be used to store the desired informationand which can accessed by computer 110. Communication media typicallyembodies computer readable instructions, data structures, programmodules or other data in a modulated data signal such as a carrier waveor other transport mechanism and includes any information deliverymedia. The term “modulated data signal” means a signal that has one ormore of its characteristics set or changed in such a manner as to encodeinformation in the signal. By way of example, and not limitation,communication media includes wired media such as a wired network ordirect-wired connection, and wireless media such as acoustic, RF,infrared and other wireless media. Combinations of the any of the aboveshould also be included within the scope of computer readable media.

The system memory 130 includes computer storage media in the form ofvolatile and/or nonvolatile memory such as read only memory (ROM) 131and random access memory (RAM) 132. A basic input/output system 133(BIOS), containing the basic routines that help to transfer informationbetween elements within computer 110, such as during start-up, istypically stored in ROM 131. RAM 132 typically contains data and/orprogram modules that are immediately accessible to and/or presentlybeing operated on by processing unit 120. By way of example, and notlimitation, FIG. 9 illustrates operating system 134, applicationprograms 135, other program modules 136, and program data 137.

The computer 110 may also include other removable/non-removable,volatile/nonvolatile computer storage media. By way of example only,FIG. 9 illustrates a hard disk drive 140 that reads from or writes tonon-removable, nonvolatile magnetic media, a magnetic disk drive 151that reads from or writes to a removable, nonvolatile magnetic disk 152,and an optical disk drive 155 that reads from or writes to a removable,nonvolatile optical disk 156 such as a CD ROM or other optical media.Other removable/non-removable, volatile/nonvolatile computer storagemedia that can be used in the exemplary operating environment include,but are not limited to, magnetic tape cassettes, flash memory cards,digital versatile disks, digital video tape, solid state RAM, solidstate ROM, and the like. The hard disk drive 141 is typically connectedto the system bus 121 through a non-removable memory interface such asinterface 140, and magnetic disk drive 151 and optical disk drive 155are typically connected to the system bus 121 by a removable memoryinterface, such as interface 150.

The drives and their associated computer storage media discussed aboveand illustrated in FIG. 9 provide storage of computer readableinstructions, data structures, program modules and other data for thecomputer 110. In FIG. 9, for example, hard disk drive 141 is illustratedas storing operating system 144, application programs 145, other programmodules 146, and program data 147. Note that these components can eitherbe the same as or different from operating system 134, applicationprograms 135, other program modules 136, and program data 137. Operatingsystem 144, application programs 145, other program modules 146, andprogram data 147 are given different numbers here to illustrate that, ata minimum, they are different copies. A user may enter commands andinformation into the computer 20 through input devices such as akeyboard 162 and pointing device 161, commonly referred to as a mouse,trackball, or touch pad. Other input devices (not shown) may include amicrophone, joystick, game pad, satellite dish, scanner, or the like.These and other input devices are often connected to the processing unit120 through a user input interface 160 that is coupled to the systembus, but may be connected by other interface and bus structures, such asa parallel port, game port or a universal serial bus (USB). A monitor191 or other type of display device is also connected to the system bus121 via an interface, such as a video interface 190. In addition to themonitor, computers may also include other peripheral output devices suchas speakers 197 and a printer 196, which may be connected through anoutput peripheral interface 190.

The computer 110 may operate in a networked environment using logicalconnections to one or more remote computers, such as a remote computer180. The remote computer 180 may be a personal computer, a server, arouter, a network PC, a peer device or other common network node, andtypically includes many or all of the elements described above relativeto the computer 110, although only a memory storage device 181 has beenillustrated in FIG. 9. The logical connections depicted in FIG. 9include a local area network (LAN) 171 and a wide area network (WAN)173, but may also include other networks. Such networking environmentsare commonplace in offices, enterprise-wide computer networks,intranets, and the Internet.

When used in a LAN networking environment, the computer 110 is connectedto the LAN 171 through a network interface or adapter 170. When used ina WAN networking environment, the computer 110 typically includes amodem 172 or other means for establishing communications over the WAN173, such as the Internet. The modem 172, which may be internal orexternal, may be connected to the system bus 121 via the user inputinterface 160, or other appropriate mechanism. In a networkedenvironment, program modules depicted relative to the computer 110, orportions thereof, may be stored in the remote memory storage device. Byway of example, and not limitation, FIG. 9 illustrates remoteapplication programs 185 as residing on memory device 181. It will beappreciated that the network connections shown are exemplary and othermeans of establishing a communications link between the computers may beused.

CONCLUSIONS

The validity of the model in extracting meaningful thermodynamicparameters in a partially reversible system is determined by thecapability of the model to determine these parameters uniquely. Thelevel of sophistication in the model imposes limitations on the results.For example, if the model has too few parameters, one may either get apoor fit or forego certain detailed description of the system. On theother hand, if there are too many parameters, a good fit may not assurea unique set of parameters and therefore render them meaningless.Although global identifiability is difficult to achieve, we checked thelocal identifiability of our model by the rigorous calculation describedabove and showed that our calculated parameter values were all uniquelydetermined.

From the results we were able to extract meaningfully relevantthermodynamic (e.g. ΔC_(P), ΔH_(m)and T_(m)) and kinetic parameters(e.g. A₁, E₁, A₂, E₂, A₃, E₃, A₄ and E₄) with varying levels ofcertainty. Two factors contribute to the uncertainty of the parameterswe obtained. The first is the uncertainty in measurements. The second isthe intrinsic sensitivity of the observables to each parameter under theconditions of the experiment. In order to estimate the uncertainties ofthe parameters in the model, we considered a 10% in proteinconcentration. Correspondingly the uncertainties induced by this errorin the parameter estimations can be determined by projecting the erroron the expectation space of theoretical observables. As a result of thequantity with the largest uncertainty is E₂, as shown in Table 1. Thisis expected since k₂ is least dependent on temperature amongst all thekinetic coefficients. Therefore in the limited temperature range of theexperiment, we could not determine the value of E₂ and A₂ very well.However, despite this difficulty, we could still determine ΔH_(m) andT_(m) relatively well. As suggested by this study, we expect improvementcould be achieved if we expanded the study to include slower and fasterDSC scan rates than those examined in the study. A more definitiveimprovement can be achieved if E₂ could be measured directly in aseparate experiment.

We believe that the model presented, though not perfect, captures themain physical processes underlying the experimental conditions tested.Several points can now be made concerning the approximations andassumptions used as they pertain to real molecular properties. The modelas presented tends to support the ΔH_(m)=E₁-E₂ expectation. It canaccount for the additional 48 kcal/mol in the total enthalpy of thepartially reversible system by allowing for populated D states (ofaggregation) on the high temperature side of the unfolding envelope.This is supported by the massive aggregation observed at temperaturesgreater than or equal to the T_(m). When k₃ and k₄ are set equal to zeroto simulate the fully reversible case based on the theory, the T_(m)(˜53.5° C.) is very near that observed experimentally for the fullyreversible case in urea (˜52.6° C.). The assumption that a combinationof first and second order reaction rates are involved in the aggregationkinetics is supported by the experimental findings that the reactionorder is ˜1.70±0.04. Finally, a reasonable prediction of aggregationrates at low temperatures (below the unfolding transition) was achievedtaking into account curvature imposed by the ΔC_(p) term used in thetheoretical treatment. There is no question that further refinementsbased on more experimental evidence will help determine the mechanismand parameters more accurately so that extrapolation to othertemperatures (above and below the T_(m)) result in meaningfulpredictions. The experimental findings suggest that stability of rhuIL1R(II) is afforded through the thermodynamic stabilization of the nativestate as suggested below the Tm, through the thermodynamic stabilizationof the native state where progress to the irreversibly denaturedaggregate is effectively blocked as in the case of the urea experiment.Furthermore, unfolded or conformationally altered protein propagate theaggregation reaction for this system.

It is anticipated that this model could be applied to better predictlevels of aggregation at low temperatures. This aspect has significantimplications with regard to fulfilling a need regarding betterestimations of shelf-life for biopharmaceuticals. Furthermore, the modelappropriately describes aggregation conditions associated with varyingconcentration factors. Hence, it is possible to run scan-rate dependentexperiments at a single concentration in the calorimeter and translatethe results into meaningful estimates that predict aggregation kineticsat other concentrations.

Having now fully described the invention, it will be appreciated bythose skilled in the art that the invention can be performed within arange of equivalents and conditions without departing from the spiritand scope of the invention and without undue experimentation. Inaddition, while the invention has been described in light of certainembodiments and examples, the inventors believe that it is capable offurther modifications. This application is intended to cover anyvariations, uses, or adaptations of the invention which follow thegeneral principles set forth above.

The specification includes recitation to the literature and thoseliterature references are herein specifically incorporated by reference.

The specification and examples are exemplary only with the particularsof the claimed invention set forth as follows:

1. A method for determining parameters for predicting aggregationkinetics of a biologically active material comprising the steps of: (a)providing measurements of conformational change of the biologicallyactive material at varying temperatures and varying times, and (b) usingthe measurements of part (a) to mathematically determine activationenergy parameters (E) and frequency factor parameters (A) associatedwith at least three different reaction rate constants, the parametersbeing predictive of aggregation kinetics of the biologically activematerial.
 2. (canceled)
 3. (canceled)
 4. (canceled)
 5. (canceled)
 6. Themethod of claim 1 wherein one or more of the following equations isused:{dot over (N)}=−k ₁ N+k ₂ U{dot over (U)}=k ₁ N−(k ₂ +k ₃)U−k ₄ U ²{dot over (D)}=k ₃ U+k ₄ U ²
 7. The method of claim 1 wherein thefollowing equation is used:${C_{P}\left( {v,T} \right)} = {{\left( {{\Delta\quad H_{m}} + {\Delta\quad{C_{P}\left( {T - T_{m}} \right)}}} \right)\left( {{- \frac{1}{v}}\overset{.}{N}} \right)} + {\Delta\quad C_{P}U} + {\frac{k_{3}}{v}\left( {E_{3} + {\Delta\quad{C_{P}^{D_{1}}\left( {T - T_{m}} \right)}}} \right)U} + {\frac{k_{4}}{V}\left( {E_{4} + {C_{P}^{D_{2}}\left( {T - T_{m}} \right)}} \right)U^{2}} + {\left( {{\Delta\quad C_{P}^{D_{1}}} + {\Delta\quad C_{P}}} \right)D_{1}} + {\left( {{\Delta\quad C_{P}^{D_{2}}} + {\Delta\quad C_{P}}} \right)D_{2}}}$8. (canceled)
 9. (canceled)
 10. (canceled)
 11. The method of claim 1wherein said determining step involves modeling aggregation, as afunction of time at different temperatures, as a first and second orderreaction.
 12. The method of claim 1, in which at least some of theparameters collectively model non-Arrhenius aspects of the aggregationkinetics.
 13. (canceled)
 14. (canceled)
 15. (canceled)
 16. (canceled)17. (canceled)
 18. The method of claim 1 further comprising one or moresteps of: determining enthalpy or free energy of transition, determiningΔCp, ΔC_(P) ^(D) ¹ , and ΔC_(P) ^(D) ² ; or determining the temperatureat which about 50% of the protein is in an unfolded state and about 50%of the protein is in its native state.
 19. (canceled)
 20. The method ofclaim 1 wherein steps (a) and (b) are carried out on a plurality ofdifferent formulations of said biologically active material. 21.(canceled)
 22. (canceled)
 23. A method for predicting aggregationkinetics of a biologically active material comprising the steps of: (a)providing activation energy parameter (E) and frequency factorparameters (A) associated with at least three different reaction rateconstants, and (b) predicting stability or aggregation kinetics as afunction of time and temperature using at least three different reactionrate constants.
 24. The method of claim 23 wherein the parameters ofstep (a) are determined by modeling aggregation, as a function of timeat different temperatures, as a first and second order reaction, and (i)using differential scanning calorimetry or size exclusion chromotographyto provide measurements of conformational change of the biologicallyactive material at varying temperatures and varying times; (ii)providing estimated activation energy and frequency factor parameters;(iii) calculating predicted measurements of conformational change basedon the estimated parameters, and (iv) using an estimation method tocompare the predicted measurements to the measurements from step (i).25. (canceled)
 26. (canceled)
 27. (canceled)
 28. The method of claim 23wherein said predicting step comprises predicting level of aggregationof said biologically active material at a temperature of 40 degrees C.or less.
 29. The method of claim 28 wherein said temperature is in arange from 4 to 25 degrees C.
 30. The method of claim 28 wherein saidtemperature is in a range from 15 to 30 degrees C.
 31. The method ofclaim 28 wherein said temperature is in a range from −5 to 15 degrees C.32. The method of claim 28 wherein said temperature is in a range from 2to 8 degrees C.
 33. The method of claim 23 wherein said predicting stepcomprises predicting level of aggregation of said biologically activematerial after a time period of three months or more.
 34. The method ofclaim 33 wherein said time period is six months or more.
 35. The methodof claim 33 wherein said time period is nine months or more.
 36. Themethod of claim 33 wherein said time period is one year or more.
 37. Themethod of claim 33 wherein said time period is two years or more. 38.The method of claim 23 wherein the predicting step comprises predictingtime to reach an unacceptable level of aggregation.
 39. The method ofclaim 38 wherein the time to reach 50% aggregation is predicted. 40.(canceled)
 41. (canceled)
 42. (canceled)
 43. The method of claim 23wherein said predicting step comprises predicting stability or level ofaggregation for a plurality of formulations of said biologically activematerial.
 44. The method of claim 43 wherein at least one of theformulations contains one or more excipients.
 45. The method of claim 43wherein at least two of the formulations are at different pH.
 46. Themethod of claim 23 wherein the effect of one or more excipients on shelflife of said biologically active material is predicted.
 47. (canceled)48. The method of claim 23 wherein said predicting step comprises usingA_(gg)(T, t)=D.
 49. (canceled)
 50. (canceled)
 51. (canceled)
 52. Acomputer-readable medium having computer-executable instructions fordetermining parameters for predicting aggregation kinetics of abiologically active material, the instructions comprising the steps of:(a) storing data of conformational change of the biologically activematerial at varying temperatures and varying times, and (b) using thedata of part (a) to mathematically determine activation energyparameters (E) and frequency factor parameters (A) associated with atleast three different reaction rate constants, the parameters beingpredictive of aggregation kinetics of the biologically active material.53. The computer-readable medium of claim 52 in which the instructionsof step (b) comprises the steps of evaluating identifiability andvariability of one or more of the parameters.
 54. The computer-readablemedium of claim 52 in which the instructions of step (b) comprisedetermining the change in heat capacity between native and denaturedstates of said biologically active material (ΔCp, ΔC_(P) ^(D) ¹ , ΔC_(P)^(D) ² ), wherein said change in heat capacity is predictive ofaggregation kinetics of the biologically active material.
 55. Thecomputer-readable medium of claim 52 in which the instructions of step(b) comprise the steps of (i) using estimated activation energy andfrequency factor parameters, (ii) calculating predicted measurements ofconformational change based on the estimated parameters, and (iii) usingan estimation method to compare predicted measurements to the data fromstep (a).
 56. The computer-readable medium of claim 55 in which theparameter estimation method is a non-linear least squares fittingmethod.
 57. (canceled)
 58. (canceled)
 59. (canceled)
 60. (canceled) 61.(canceled)
 62. (canceled)
 63. (canceled)
 64. The computer-readablemedium of claim 52 in which the data includes measurements ofconformational change of the biologically active material underconditions that result in significant irreversible unfolding.
 65. Thecomputer-readable medium of claim 64 in which the data comprises dataobtained from differential scanning calorimetry.
 66. Thecomputer-readable medium of claim 64 in which the data comprises dataobtained from size exclusion chromatography.
 67. The computer-readablemedium of claim 52 in which the data comprises data of conformationalchange of the biologically active material measured as a function oftemperature varied uniformly over time.
 68. The computer-readable mediumof claims 67 in which the instructions include applying a weightingfactor dependent on the scan rate.
 69. (canceled)
 70. (canceled) 71.(canceled)
 72. (canceled)
 73. (canceled)
 74. A computer readable mediumhaving computer-executable instructions for predicting aggregationkinetics of a biologically active material comprising the steps of: (a)storing activation energy parameter (E) and frequency factor parameters(A) associated with at least three different reaction rate constants,and (b) predicting stability or aggregation kinetics as a function oftime and temperature using at least three different reaction rateconstants.
 75. The computer readable medium of claim 74 in which theparameters of step (a) have been obtained by modeling aggregation, as afunction of time at different temperatures, as a first and second orderreaction, and (i) using differential scanning calorimetry or sizeexclusion chromotography to provide measurements of conformationalchange of the biologically active material at varying temperatures andvarying times; (ii) providing estimated activation energy and frequencyfactor parameters; (iii) calculating predicted measurements ofconformational change based on the estimated parameters, and (iv) usingan estimation method to compare the predicted measurements to themeasurements from step (i).
 76. The computer readable medium of claim 74wherein activation energy parameters (E) and frequency factor parameters(A) associated with at least four reaction rate constants are stored.77. The computer-readable medium of claim 74 wherein activation energyparameters (E) and frequency factor parameters (A) associated with nomore than four reaction rate constants are stored.
 78. Thecomputer-readable medium of claim 23 wherein said predicting step (b)comprises predicting level of aggregation of said biologically activematerial as a function of temperature, time and concentration of saidbiologically active material.
 79. (canceled)
 80. (canceled) 81.(canceled)
 82. (canceled)
 83. (canceled)
 84. (canceled)
 85. (canceled)86. (canceled)
 87. (canceled)
 88. (canceled)
 89. (canceled) 90.(canceled)
 91. The computer-readable medium of claim 74 wherein saidpredicting step comprises predicting aggregation half-life of saidbiologically active material as a function of temperature andconcentration of said biologically active material.
 92. Thecomputer-readable medium of claim 74 wherein said predicting stepcomprises predicting shelf-life of said biologically active material atone or more storage temperatures.
 93. The computer-readable medium ofclaim 74 wherein said predicting step comprises predicting an optimalstorage temperature.
 94. (canceled)
 95. (canceled)
 96. (canceled) 97.(canceled)
 98. The computer-readable medium of claim 74 furthercomprising instructions for selecting an optimal formulation. 99.(canceled)
 100. The computer-readable medium of claim 52 wherein thedata for the biologically active material is data for a protein. 101.The computer-readable medium of claim 100 wherein the data is for ahormone, cytokine, hematopoietic factor, growth factor, antibody,antiobesity factor, trophic factor, anti-inflammatory factor, antibodyor enzyme.
 102. The computer-readable medium of claim 101 wherein thedata is for erythropoietin, granulocyte-colony stimulating factor, stemcell factor, or leptin.
 103. A computing apparatus, comprising: adisplay unit that is capable of generating video images; an inputdevice; a processing apparatus operatively coupled to said display unitand said input device, said processing apparatus comprising a processorand a memory operatively coupled to said processor; the processingapparatus being programmed to determine parameters for predictingaggregation kinetics of a biologically active material by performingsteps comprising: (a) storing measurements of conformational change ofthe biologically active material at varying temperatures and varyingtimes, and (b) using the measurements of part (a) to determineactivation energy parameters (E) and frequency factor parameters (A)associated with at least three different reaction rate constants, theparameters being predictive of aggregation kinetics of the biologicallyactive material.
 104. The computing apparatus of claim 103, in which theprocessing apparatus is programmed with the steps of: (i) receivinginput of differential scanning calorimetry or size exclusionchromotography measurements of conformational change of the biologicallyactive material at varying temperatures and varying times; (ii)receiving input of estimated activation energy and frequency factorparameters; (iii) calculating predicted measurements of conformationalchange based on the estimated parameters, and (iv) using an estimationmethod to compare the predicted measurements to the measurements fromstep (i).
 105. A computing apparatus, comprising: a display unit that iscapable of generating video images; an input device; a processingapparatus operatively coupled to said display unit and said inputdevice, said processing apparatus comprising a processor and a memoryoperatively coupled to said processor; the processing apparatus beingprogrammed to predict aggregation kinetics of a biologically activematerial by performing steps comprising: (a) storing activation energyparameter (E) and frequency factor parameters (A) associated with atleast three different reaction rate constants, and (b) predictingstability or aggregation kinetics as a function of time and temperatureusing at least three different reaction rate constants.
 106. Thecomputing apparatus of claim 105, in which processing apparatus isprogrammed with the steps of: (i) receiving input of differentialscanning calorimetry or size exclusion chromotography measurements ofconformational change of the biologically active material at varyingtemperatures and varying times; (ii) receiving input of estimatedactivation energy and frequency factor parameters; (iii) calculatingpredicted measurements of conformational change based on the estimatedparameters; and (iv) using an estimation method to compare the predictedmeasurements to the measurements from step (i).